TMUA Sequences and Series — Practice Questions by Topic

These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.

MM2Sequences and series
147 questions103 on Paper 1 · 44 on Paper 2
9 free solutionsThe rest show the correct letter only

Covers: arithmetic and geometric sequences, recurrence relations, sums of series, including sums to infinity, quadratic sequences, finding a later term from the first few.

Paper 1 (103)Paper 2 (44)Practise this topic

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Question 1Sequences and series

In the expansion of (1 + x)ⁿ, where n is a positive integer, the coefficients of x^k, x^k+1 and x^k+2 are in the ratio 1 : 2 : 3. What is n?

What this topic tests

Sequences and series here means arithmetic and geometric sequences, recurrence relations, sums including a sum to infinity, quadratic sequences, and finding a later term from the first few.

How it is assessed

This is Section 1, on both papers in this set. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.

This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.

Key methods

Arithmetic and geometric

An arithmetic sequence adds a constant difference. A geometric sequence multiplies by a constant ratio. The sum of an infinite geometric series needs the absolute value of that ratio to be less than 1.

A recurrence is not yet a formula

u(n+1) in terms of u(n) tells you how to step forward. A closed form, when the question asks for one, has to be checked against the first term and against the step.

Common mistakes

Using the partial sum as the term

The sum of the first n terms and the nth term are different. A formula for one does not answer a question about the other.

An infinite sum that does not converge

If the common ratio is 1 or more, or −1 or less, there is no sum to infinity. Reporting a/(1 − r) anyway is the usual wrong option.

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

The 5th term of an arithmetic sequence is 18 and the 11th term is 48. Find the 16th term.

A. 68
B. 73
C. 78
D. 93
E. 98

Answer: B. 73

Worked solution. The common difference is (48 − 18)/(11 − 5) = 30/6 = 5. From the 11th term to the 16th term is 5 steps, so the 16th term is 48 + 5 × 5 = 73.

Why the other options look right. A adds 10 differences to the 5th term, using 16 − 5 − 1 steps, and gets 18 + 50 = 68. C steps from the 11th term by 6 differences instead of 5, giving 48 + 30 = 78. D treats 18 as the first term and computes 18 + 15 × 5 = 93. E adds 16 differences to the 5th term, giving 18 + 80 = 98.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

Q1. In the expansion of (1 + x)ⁿ, where n is a positive integer, the coefficients of x^k, x^k+1 and x^k+2 are in the ratio 1 : 2 : 3. What is n?
Free · worked solution included
Answer and worked solution
Answer: D. 14
Worked solution. The coefficient of x^r is C(n, r), and C(n, r + 1)/C(n, r) = (n − r)/(r + 1). The first ratio gives (n − k)/(k + 1) = 2, so n = 3k + 2. The second ratio gives C(n, k + 2)/C(n, k + 1) = (n − k − 1)/(k + 2) = 3/2, so 2n − 2k − 2 = 3k + 6 and 2n = 5k + 8. Substituting n = 3k + 2 gives 6k + 4 = 5k + 8, so k = 4 and n = 14. Check: C(14, 4) = 1001, C(14, 5) = 2002 and C(14, 6) = 3003, which are in the ratio 1 : 2 : 3.
Why the other options look right. A reports k = 4, the power of x in the first of the three terms, instead of n. B reports k + 2 = 6, the power of x in the third term, instead of n. C uses n − k instead of n − k − 1 in the second ratio, so (n − k)/(k + 2) = 3/2 together with n = 3k + 2 gives k = 2 and n = 8. E writes the ratio C(n, r + 1)/C(n, r) as (n − r)/r instead of (n − r)/(r + 1), so (n − k)/k = 2 and (n − k − 1)/(k + 1) = 3/2 give k = 5 and n = 15.
Q2. The 5th term of an arithmetic sequence is 18 and the 11th term is 48. Find the 16th term.
Free · worked solution included
Answer and worked solution
Answer: B. 73
Worked solution. The common difference is (48 − 18)/(11 − 5) = 30/6 = 5. From the 11th term to the 16th term is 5 steps, so the 16th term is 48 + 5 × 5 = 73.
Why the other options look right. A adds 10 differences to the 5th term, using 16 − 5 − 1 steps, and gets 18 + 50 = 68. C steps from the 11th term by 6 differences instead of 5, giving 48 + 30 = 78. D treats 18 as the first term and computes 18 + 15 × 5 = 93. E adds 16 differences to the 5th term, giving 18 + 80 = 98.
Q3. Evaluate the sum from k = 1 to 7 of (3k + 2).
Free · worked solution included
Answer and worked solution
Answer: D. 98
Worked solution. Split the sum into 3 times the sum of the first 7 positive integers, plus 2 added 7 times. The triangular sum is 7 × 8 / 2 = 28, so the total is 3 × 28 + 14 = 84 + 14 = 98.
Why the other options look right. A sums 3k from k = 1 to 7 and forgets the +2 in each term, leaving 84. B adds that constant only once, giving 86. C takes the first term to be 3, so (7/2)(3 + 23) = 91. E uses 8 terms in the series formula, (8/2)(5 + 23) = 112.
Q4. A geometric sequence has third term 54 and sixth term −2. What is the first term?
Free · worked solution included
Answer and worked solution
Answer: C. 486
Worked solution. Write the terms as ar² = 54 and ar⁵ = −2. Dividing, r³ = −2/54 = −1/27, so r = −1/3. Then a = 54/r² = 54 × 9 = 486.
Why the other options look right. A takes the ratio to be −1/27 without taking a cube root, so it divides 54 by (1/27)² = 1/729 and gets 39366. B divides 54 by r³ = −1/27 instead of by r², giving −1458. D divides 54 by r only once, giving −162. E reports the third term instead of the first.
Q5. The sum of the first n terms of an arithmetic sequence is S_n = 5n² − 2n. What is the common difference?
Free · worked solution included
Answer and worked solution
Answer: D. 10
Worked solution. Here S₁ = 5 − 2 = 3 and S₂ = 20 − 4 = 16, so the second term is 16 − 3 = 13 and the difference is 13 − 3 = 10. In general S_n = (d/2) n² + (a − d/2) n, so d/2 = 5 and d = 10.
Why the other options look right. A reads off the coefficient of n, which is −2. B reports the first term S₁ = 5 − 2 = 3 instead of the common difference. C takes the leading coefficient 5 to be the common difference, forgetting the factor 1/2 in S_n = (d/2)n² + …. E finds a₃ = S₃ − S₂ = 39 − 16 = 23 and subtracts the first term, a₃ − a₁ = 23 − 3 = 20, forgetting that this spans two common differences.
Q6. Find the coefficient of x² in the expansion of (3 − 2x)⁵.
Free · worked solution included
Answer and worked solution
Answer: D. 1080
Worked solution. The term in x² is C(5, 2) × 3³ × (−2x)². That is 10 × 27 × 4 x² = 1080 x², so the coefficient is 1080.
Why the other options look right. A multiplies by 2 rather than by (−2)², giving 10 × 27 × 2 = 540. B treats (−2)² as −4 and obtains −1080. C uses 3² instead of 3³ for the power of 3, giving 10 × 9 × 4 = 360. E omits the factor (−2)² and stops at C(5, 2) × 3³ = 270.
Q7. The nth term of a sequence is 1/[n(n + 2)]. Find the sum of its first 10 terms.
Free · worked solution included
Answer and worked solution
Answer: C. 175/264
Worked solution. Use 1/[n(n + 2)] = (1/2)(1/n − 1/(n + 2)). Therefore the sum from n = 1 to 10 is (1/2)[1 + 1/2 − 1/11 − 1/12], because all the terms from 1/3 through 1/10 cancel. The expression in brackets is 3/2 − 23/132 = 175/132, so the required sum is 175/264.
Why the other options look right. A obtains the correct bracket 175/132 but omits the factor 1/2 from the partial-fraction identity. B keeps the factor 1/2 but telescopes as if consecutive terms cancelled, using 1/n − 1/(n + 1) in place of 1/n − 1/(n + 2), which leaves (1/2)(1 − 1/11) = 5/11. D replaces n + 2 by n + 1 and telescopes the different sum from 1 to 10 of 1/[n(n + 1)], giving 10/11. E keeps the two initial positive terms but drops the uncancelled final terms, giving (1/2)(1 + 1/2) = 3/4.
Q8. The sequence b_n satisfies b_n = 5 b_n−1 − 6 b_n−2 for every integer n ≥ 3, with b₁ = 1 and b₂ = 4. Which expression equals b_n for every positive integer n?
Free · worked solution included
Answer and worked solution
Answer: B. 2 × 3ⁿ⁻¹ − 2ⁿ⁻¹
Worked solution. The characteristic equation is r² − 5r + 6 = 0, so (r − 2)(r − 3) = 0. Thus b_n = p × 2ⁿ + q × 3ⁿ. The initial conditions give 2p + 3q = 1 and 4p + 9q = 4. Subtracting twice the first equation from the second leaves 3q = 2, so q = 2/3 and p = −1/2. Therefore b_n = −(1/2) × 2ⁿ + (2/3) × 3ⁿ = 2 × 3ⁿ⁻¹ − 2ⁿ⁻¹. This matches b₁ = 1 and b₂ = 4, and the next term 5 × 4 − 6 × 1 = 14 agrees with the formula.
Why the other options look right. A uses coefficient 1 on both 3ⁿ and 2ⁿ, which gives b₂ = 5 rather than 4. C treats the given values as b₀ and b₁, producing 2 × 3ⁿ − 2ⁿ, and that expression equals 4 when n = 1. D has the correct two magnitudes but adds them, so it equals 3 when n = 1. E adds 3ⁿ⁻¹ and 2ⁿ⁻¹ with coefficient 1, so it equals 2 when n = 1.
Q9. For each positive integer n, let T_n = 1 × 1 + 2 × 3 + 3 × 5 + … + n(2n − 1). Which expression is equal to T_n?
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Answer
Answer: E. n(n + 1)(4n − 1)/6
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Q10. A sequence of points is defined by P₁ = (3, 1) and by the rule that P_n+1 is the image of P_n after a rotation of 90 degrees anticlockwise about the origin. What is P₇?
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Answer
Answer: D. (−3, −1)
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Q11. A sequence is defined by a₁ = 1 and a_n+1 = 2 a_n + 3. What is a₆?
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Answer
Answer: C. 125
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Q12. An arithmetic sequence has first term 5 and 40th term 83. What is the sum of its first 40 terms?
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Answer
Answer: C. 1760
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Q13. A geometric sequence has positive terms. Its 2nd term is 3 and its 6th term is 48. What is the common ratio?
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Answer
Answer: A. 2
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Q14. What is the coefficient of x⁴ in the expansion of (x³ + 2x + 5)³?
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Answer
Answer: C. 60
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Q15. The positive integers are written in rows, with row k containing the next k integers. Row 1 is 1, row 2 is 2, 3, row 3 is 4, 5, 6, and so on. From each row, keep the entries whose parity matches the row number: odd entries from odd rows and even entries from even rows. List the kept numbers in increasing order as a₁, a₂, a₃, …. If a_n = 50, what is n?
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Answer
Answer: D. 26
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Q16. Evaluate the sum from n = 1 to n = 23 of (−1)ⁿ⁺² + (−1)ⁿ⁺⁴ + (−1)ⁿ⁺⁶ − (−1)ⁿ⁻¹.
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Answer
Answer: B. −4
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Q17. Two geometric series each have first term 6, and both converge. Adding corresponding terms produces a series S whose first three terms are 12, 5 and 13/6. Find the sum to infinity of S.
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Answer
Answer: E. 21
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Q18. The first three terms of an arithmetic progression are 2p, q and p², where p < 0. The first three terms of a geometric progression are 2p, p² and q. Find the sum of the first 8 terms of the arithmetic progression.
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Answer
Answer: B. 26
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Q19. The sequence x_n is defined by x₁ = 3 and x_n+1 = (x_n + 3)/(1 − x_n). The first three terms are 3, −3 and 0. What is the value of x₉₉?
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Answer
Answer: A. 0
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Q20. In the expansion of (a + bx)⁴, the coefficient of x³ is 9 times the coefficient of x. Given that a and b are positive integers, what is the smallest possible value of a + b?
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Answer
Answer: B. 4
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Q21. For each positive integer n, F(n) is (1/n) times the integral from 0 to n of (3n − 3x) dx, and G(n) is the sum of F(1) + F(2) + ... + F(n). What is the smallest positive integer n such that G(n) > 90?
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Answer
Answer: C. 11
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Q22. In an arithmetic progression, the sum of the first 10 terms is 50 and the sum of the next 10 terms (terms 11 to 20) is 250. What is the sum of terms 21 to 30?
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Answer
Answer: C. 450
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Q23. The non-zero constant k is chosen so that the coefficients of x⁴ in the expansions of (1 + kx²)⁵ and (k + x)⁷ are equal. What is k?
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Answer
Answer: B. 2/7
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Q24. A sequence is defined by u₁ = 2 and u_n+1 = 3u_n + 1 for n ≥ 1. Which expression gives u_n?
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Answer
Answer: A. (5 × 3ⁿ⁻¹ − 1)/2
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Q25. Find the sum to infinity of the series 1 + 3/2 + 5/4 + 7/8 + 9/16 + …, whose numerators are the odd numbers 1, 3, 5, … and whose denominators are the powers of 2.
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Answer
Answer: C. 6
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Q26. The sequence is given by x₁ = 5 and x_n+1 = √(x_n) for n ≥ 1. What is x₄₀?
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Answer
Answer: A. 5^2⁻³⁹
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Q27. A sequence is defined by u₁ = 3 and u_n+1 = 2u_n − 1. What is the sum of its first six terms?
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Answer
Answer: A. 132
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Q28. The 1st, 2nd and 3rd terms of a geometric progression are also the 1st, 3rd and 4th terms, respectively, of an arithmetic progression. The sum to infinity of the geometric progression is 18. What is its first term?
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Answer
Answer: B. 9
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Q29. Evaluate the sum from k = 1 to k = 20 of 1/[k(k + 1)].
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Answer
Answer: A. 20/21
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Q30. A sequence satisfies x_n+1 = (x_n + p)/(x_n + q), where p and q are real constants. Given x₁ = 2, x₂ = 5 and x₃ = 8, find x₄.
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Answer
Answer: E. 29
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Q31. A geometric sequence has positive integer first term and integer common ratio r > 1. Its partial sums satisfy S₁₂ = k S₄ for a positive integer k. What is the smallest possible k?
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Answer
Answer: E. 273
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Q32. Evaluate the sum from n = 0 to infinity of sin(nπ + π/6) / 3ⁿ.
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Answer
Answer: C. 3/8
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Q33. How many of the nine coefficients in the expansion of (1 + 2x)⁸ are multiples of 32?
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Answer
Answer: D. 6
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Q34. The integer k is chosen at random from the 20 non-zero integers from −10 to 10 inclusive. The geometric series S = 1 + 2/k + 4/k² + 8/k³ + … has common ratio 2/k. What is the probability that S is a finite number less than 3/2?
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Answer
Answer: C. 3/5
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Q35. Find the value of 1² − 2² + 3² − 4² + ... + 99² − 100².
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Answer
Answer: A. −5050
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Q36. What is the coefficient of x² in (3 − x²)[(1 + x + 2x²)⁴ − (1 + 5x³)³]?
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Answer
Answer: C. 42
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Q37. The expansion of (2 + ax)⁴ is 16 + px + 216x² + qx³ + rx⁴, where a, p, q and r are positive real numbers. Find p + q + r.
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Answer
Answer: D. 393
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Q38. What is the value of 1 × 2 + 2 × 2² + 3 × 2³ + … + 10 × 2¹⁰?
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Answer
Answer: E. 18434
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Q39. The inequality (n + 3) + (2n + 6) + (3n + 9) + … + (20n + 60) > k holds for every integer n ≥ 0. Which statement must be true?
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Answer
Answer: B. k < 630
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Q40. A geometric series has first term 6 and common ratio r, with −1 < r < 1. The sum to infinity of the cubes of its terms is 3/7 of the cube of its own sum to infinity. Find r.
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Answer
Answer: C. 1/4
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Q41. A geometric sequence has first term 3 and common ratio 2. What is the 5th term?
Free · worked solution included
Answer and worked solution
Answer: C. 48
Worked solution. The nth term is 3 × 2^{n − 1}. For n = 5 this is 3 × 2⁴ = 3 × 16 = 48.
Why the other options look right. A is the 4th term, 3 × 2³ = 24, from stopping one term early. B is 3 × 2 × 5 = 30, multiplying by the ratio times the term number instead of using a power of the ratio. D is the 6th term, 3 × 2⁵ = 96, from using 2ⁿ instead of 2^{n − 1}. E is 2 × 3⁴ = 162, swapping the roles of the first term and the ratio.
Q42. The coefficient of x³ in the expansion of (2 + x + x²)⁵ is twice the coefficient of x² in the expansion of (2 + ax)⁵. Find every possible value of the constant a.
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Answer
Answer: C. ±(√5)/2
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Q43. The coefficient of x³ in the expansion of (3 + cx)⁵ is three times the coefficient of x² in the expansion of (1 + cx)⁶. Given that c is nonzero, find c.
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Answer
Answer: A. 1/2
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Q44. A geometric sequence with real common ratio has second term 6 and fifth term −48. What is the sum of its first six terms?
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Answer
Answer: D. 63
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Q45. When (1 + x − x²)(1 + 3x)⁴ is expanded, what is the coefficient of x²?
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Answer
Answer: C. 65
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Q46. A geometric sequence has real common ratio. Its second term is 4, and the sum of its second, third and fourth terms is 52. What is the sum of all the possible values of its first term?
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Answer
Answer: D. 1/3
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Q47. A geometric series has first term 6 and sum to infinity 10. Find the common ratio.
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Answer
Answer: E. 2/5
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Q48. The first three terms of an arithmetic sequence are k − 1, 2k + 1 and 4k − 1, where k is a constant. What is the sum of the first 10 terms of the sequence?
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Answer
Answer: A. 300
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Q49. When (2 + x − x²)(1 − 3x)³ is expanded, what is the coefficient of x²?
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Answer
Answer: D. 44
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Q50. A geometric sequence has second term 8 and fifth term −27. Find the first term.
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Answer
Answer: B. −16/3
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Q51. A geometric series has first term 6 and sum to infinity 4. Find the common ratio.
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Answer
Answer: E. −1/2
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Q52. The arithmetic series 69 + 65 + 61 + ... is added up one term at a time. What is the greatest value reached by the running total?
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Answer
Answer: C. 630
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Q53. When (3 − x + 2x²)(2 + x)³ is expanded in powers of x, what is the coefficient of x²?
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Answer
Answer: C. 22
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Q54. A geometric series has sum to infinity 27, and the sum of its first three terms is 26. Find the first term.
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Answer
Answer: D. 18
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Q55. A geometric series has first term 4 and sum to infinity 5. Find the common ratio.
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Answer
Answer: D. 1/5
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Q56. What is the sum of all the integers between 1 and 200 that leave remainder 2 when divided by 7?
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Answer
Answer: B. 2900
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Q57. Find the coefficient of x² in the expansion of (3 + x)(2 − x)⁴.
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Answer
Answer: A. 40
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Q58. A geometric sequence has second term 3 and fifth term 192. Find its first term.
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Answer
Answer: E. 3/4
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Q59. An infinite geometric series has first term 9 and sum 45. Find the common ratio.
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Answer
Answer: B. 4/5
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Q60. Find the sum of all the odd integers between 100 and 200.
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Answer
Answer: D. 7500
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Q61. In the product (2 + 3x + x²)(1 + x)⁵, what is the coefficient of x³?
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Answer
Answer: D. 55
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Q62. For every positive integer n, the sum of the first n terms of a sequence is 3ⁿ⁺¹ + k, where k is a constant. The sequence is geometric. What is k?
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Answer
Answer: C. −3
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Q63. The sequence has nth term u_n = 1/((2n − 1)(2n + 1)). Find the sum of its first 10 terms.
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Answer
Answer: B. 10/21
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Q64. An arithmetic sequence has 3rd term 11 and 10th term 39. What is the sum of its first 20 terms?
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Answer
Answer: E. 820
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Q65. When (2 − x + 3x²)(1 + x)⁵ is expanded, what is the coefficient of x²?
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Answer
Answer: C. 18
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Q66. A geometric sequence has third term 48 and sixth term 6. Find its first term.
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Answer
Answer: B. 192
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Q67. An infinite geometric series has first term 15 and sums to 9. Find the common ratio.
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Answer
Answer: C. −2/3
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Q68. What is the sum of all positive integers less than 100 that are multiples of 4 or of 6?
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Answer
Answer: A. 1584
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Q69. When (1 + 2x − 3x²)(1 + 4x)³ is expanded, what is the coefficient of x²?
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Answer
Answer: C. 69
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Q70. In a geometric sequence the third term is 50 and the sixth term is −6250. What is the first term?
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Answer: B. 2
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Q71. A geometric series has first term 8 and sum to infinity 5. What is the common ratio?
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Answer: A. −3/5
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Q72. The nth term of a sequence is 2ⁿ + 3n. What is the sum of the first 10 terms of the sequence?
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Answer
Answer: D. 2211
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Q73. A sequence is given by a_n = (n + 2)/(n + 5). What is the product of the first 30 terms, starting at n = 1?
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Answer: D. 2/1309
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Q74. The sum of the first n terms of a sequence is S_n = 3n² − n. What is the 10th term?
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Answer: C. 56
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Q75. An arithmetic progression has first term a, where a ≠ 0, and common difference d. The sum of its first 12 terms is three times the sum of its first 6 terms. What is d/a?
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Answer: A. 2/7
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Q76. A sequence is defined by u₀ = 1 and, for every integer n ≥ 1, u_n equals twice the sum of u₀ through u_n−1. Find the sum from r = 0 to infinity of 1/u_r.
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Answer: D. 7/4
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Q77. Find the coefficient of x⁶ in the expansion of (1 + x + x²)⁷.
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Answer: E. 357
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Q78. How many solutions does the equation cos(2x) + cos²(2x) + cos³(2x) + ... = 1 have in the interval 0 ≤ x < 2π?
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Answer: C. 4
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Q79. What is the coefficient of x² in the expansion of (1 + x)(1 + 2x)(1 + 3x) ... (1 + 10x)?
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Answer: B. 1320
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Q80. The sum 1 + 4 + 7 + ... + (3n − 2) of the first n terms of an arithmetic sequence is greater than 1001. What is the smallest possible value of the positive integer n?
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Answer: C. 27
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Q81. A geometric series has first term a ≠ 0 and common ratio r, with −1 < r < 1. A second geometric series has the squares of those terms as its terms. The sum to infinity of the second series is one third of the square of the sum to infinity of the first. What is r?
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Answer: E. 1/2
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Q82. A geometric sequence has first term a and common ratio r, with a not equal to 0 and r > 1. The sum of its first 6 terms is three times the sum of its first 3 terms. What is r?
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Answer: B. 2^1/3
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Q83. Let floor(t) denote the greatest integer less than or equal to t. The sequence is defined by a_n = floor(√n). What is the sum of the first 48 terms, from n = 1 to n = 48?
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Answer: D. 203
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Q84. A positive number k is chosen so that k + 2, k² + k and k² + 5k + 3 are the first three terms of an arithmetic sequence. What is the 11th term of that sequence?
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Answer: B. 237
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Q85. A sequence begins with u(1) = 3 and satisfies u(n + 1) − u(n) = 2n + 1 for every positive integer n. Find u(10).
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Answer: B. 102
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Q86. A geometric series with first term a and common ratio r has sum to infinity 100. Another geometric series with the same first term, but with common ratio r/3, has sum to infinity 60. Find a.
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Answer: C. 50
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Q87. An arithmetic sequence satisfies u₃ + u₈ + u₁₃ = 51 and u₅ × u₁₁ = 280. What is the greatest possible value of u₁?
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Answer: D. 24
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Q88. The nonzero numbers p, q and s are the first three terms of a geometric sequence. The numbers p, 3q and s form an arithmetic sequence. Find every possible common ratio of the geometric sequence.
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Answer: C. 3 ± 2√2
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Q89. A geometric sequence of real numbers satisfies u₂ × u₉ = 8. What is the product of the first 10 terms, u₁ × u₂ × ... × u₁₀?
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Answer: A. 2¹⁵
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Q90. A sequence is defined by b₀ = 2 and, for every integer n ≥ 1, b_n = b₀ × b₁ × ... × b_n−1 + 1. Find the sum of the reciprocals 1/b_r from r = 0 to infinity.
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Answer: C. 1
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Q91. A sequence satisfies y₁ = 1, y₂ = 4 and y_n+1 = (1 + y_n) / y_n−1 for every integer n ≥ 2. What is y₂₀₂₅?
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Answer
Answer: D. 1/2
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Q92. A series begins 1 + 1/3 + 1/4 + 1/12 + 1/16 + 1/48 + ... . The terms in odd positions form a geometric progression with first term 1, and the terms in even positions form a geometric progression with first term 1/3. Both progressions have common ratio 1/4. What is the sum of the first 2n terms, where n is a positive integer?
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Answer: C. (16/9)(1 − 1/4ⁿ)
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Q93. For each integer n ≥ 1, let u_n = (−1)ⁿ⁺¹ × 3n, and let w_n be the sum of u₁ through u_n. Find the value of n for which w_n = 750.
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Answer: B. 499
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Q94. A sequence is defined by a₁ = x and a_n+1 = −1 − 1/(2a_n) for every integer n ≥ 1, where x is chosen so that every term is defined. What is the smallest positive integer p such that, for every such choice of x, a_n+p = a_n for every n?
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Answer: C. 4
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Q95. A sequence is defined by a₁ = 1 and a_n+1 = (a_n + k)/(1 − k a_n) for every integer n ≥ 1, where k is a nonzero real constant chosen so that every term is defined. For which values of k does the sequence have minimal period 3?
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Answer
Answer: C. √3 or −√3
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Q96. What is the coefficient of x² in the expansion of (2 + x)[(1 − 2x + 3x²)⁴ − (1 + x³)⁵]?
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Answer: D. 64
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Q97. What is the coefficient of x² in the expansion of (1 − x + 4x²)⁶?
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Answer: E. 39
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Q98. In the expansion of (2 + x)⁷, what is the sum of the coefficients of the even powers of x, including the constant term?
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Answer: D. 1094
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Q99. What is the coefficient of x⁴ in the expansion of (1 − x)³ (1 + x)⁷?
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Answer: D. −14
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Q100. A sequence satisfies u_n = 3u_n−1 + 2u_n−2 for every integer n ≥ 3, with u₁ = 1 and u₂ = 5. What is the limit of u_k / u_k−1 as k tends to infinity?
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Answer: D. (3 + √17)/2
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Q101. For |x| < 1, the series p(x) = −4 + x + x² + x³ + ... equals 0. What is x?
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Answer: B. 4/5
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Q102. Evaluate the sum from n = 2 to 10 of 1/(n² − 1).
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Answer: C. 36/55
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Q103. Evaluate 1 + 2 × (1/2) + 3 × (1/2)² + 4 × (1/2)³.
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Answer: D. 13/4
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Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

Q1. The constant term in the expansion of (3x + a/x)⁴ is 216, and a is positive. What is a?
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Answer
Answer: B. 2
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Q2. The sequences a_k and b_k are arithmetic, and a_k/b_k has the same value for k = 1, 2, 3 and 4. Given a₁ = 180, a₄ = 72 and b₁ = 90, what is b₂?
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Answer: C. 72
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Q3. A sequence is defined by a₁ = 2 and a_n+1 = a_n/(1 + a_n) for n ≥ 1. What is a₅₀?
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Answer
Answer: E. 2/99
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Q4. The sequence a_n is geometric with real terms, a₃ = −2 and a₇ = −162. What is a₅?
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Answer
Answer: E. −18
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Q5. An arithmetic sequence has first term a and common difference d, where a and d are positive integers. For some integer n ≥ 3, the sum of the first n terms is 100. How many different triples (a, d, n) are possible?
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Answer: B. 20
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Q6. The first term of a geometric progression is 3√2 and the fourth term is 3/2. What is the sum to infinity of this geometric progression?
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Answer: D. 6(1 + √2)
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Q7. A sequence u₀, u₁, u₂, ... is defined by u₀ = 1 and, for n ≥ 1, u_n equals the integral from 0 to 1 of 6x u_n−1 dx. What is the value of u₁₂?
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Answer
Answer: B. 3¹²
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Q8. The function f is defined on the positive integers by f(1) = 15, and for n ≥ 1: f(n + 1) = f(n) + 9 if f(n) is odd, and f(n + 1) = f(n)/2 if f(n) is even. The function g is defined by g(1) = 7, and for n ≥ 1: g(n + 1) = g(n) + 1 if g(n) is odd, and g(n + 1) = g(n)/2 if g(n) is even. What is f(100) − g(100)?
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Answer
Answer: C. 4
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Q9. Sequence 1 is an arithmetic progression with first term 7 and common difference 4. Sequence 2 is an arithmetic progression with first term 3 and common difference 6. Let N be the 20th number that appears in both sequences. What is the remainder when N is divided by 11?
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Answer: B. 1
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Q10. Every term of the infinite geometric progression u₁, u₂, u₃, ... is a positive integer, and its common ratio is r. Which of the following must be true? I: r is an integer. II: If u₁ is odd, then every term is odd. III: The terms u₁, u₃, u₅, ... form a geometric progression whose common ratio is a perfect square.
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Answer: D. I and III only
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Q11. Three distinct positive integers form an increasing geometric sequence, and their sum is 91. What is the largest possible value of the first term?
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Answer: B. 25
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Q12. The first term a and the common difference d of an arithmetic sequence are non-zero integers. For some positive integer m, the sum of the first m terms is equal to the sum of the first 3m terms. Which of the following must be true? I: d is even. II: a and d have opposite signs. III: d is a factor of a.
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Answer: C. I and II only
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Q13. Find the coefficient of x⁵ in (2 + x)⁵ (1 + x + x² + x³ + x⁴ + x⁵).
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Answer: D. 243
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Q14. A selection of n terms is taken from the arithmetic progression 2, 5, 8, ..., 62. What is the smallest n that forces two distinct selected terms to add to 64?
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Answer: C. 12
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Q15. A sequence is defined by u₁ = a, u₂ = b and u_n+2 = u_n+1 − u_n for n ≥ 1, where a and b are positive integers. Which of these statements must be true? I: u₂₀ = b. II: u₁ + u₂ + u₃ + u₄ + u₅ + u₆ = 0. III: u₂₀₃ is positive.
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Answer: D. I and II only
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Q16. The sequence a_n satisfies a₁ = 3 and a_n + a_n+1 = 4n for every integer n ≥ 1. What is a₁ + a₂ + a₃ + ... + a₄₁?
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Answer: E. 1683
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Q17. Start with the number 1. At each step, if the current value is a multiple of 3, replace it by one third of itself; otherwise add 4 to it. The process is run until 99 values have been produced, counting the start as the first value. What is the sum of these 99 values?
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Answer: C. 447
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Q18. The function G is defined on the positive integers by G(1) = 2, and for n ≥ 2: G(n) = 3G(n − 1) + 1 if 2 divides n but 3 does not; G(n) = 2G(n − 1) − 1 if 3 divides n but 2 does not; G(n) = G(n − 1) + 4 if both 2 and 3 divide n; G(n) = G(n − 1) if neither 2 nor 3 divides n. What is G(12)?
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Answer
Answer: B. 800
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Q19. A sequence starts at 4. Each following term is 8 divided by the square of the term before it. The 10th term can be written as 2 to the power a. What is a?
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Answer: B. −511
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Q20. A sequence has t₁ = 2 and t₂ = 3, and for n ≥ 2, t_n+1 = (1 + t_n)/t_n−1. What is t₁ + t₂ + ... + t₂₀₂₄?
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Answer
Answer: C. 3644
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Q21. The function f is defined on the positive integers by f(1) = 5. For n ≥ 1, f(n + 1) = 3f(n) − 1 when f(n) is odd, and f(n + 1) = f(n)/2 when f(n) is even. Find the sum f(1) + f(2) + ... + f(48).
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Answer
Answer: E. 530
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Q22. The function f is defined on the positive integers by f(1) = −1 and, for every integer m ≥ 1, f(2m) = (f(m))^{m + 1} and f(2m + 1) = (f(m))^m. Find f(1) + f(2) + ... + f(60).
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Answer: B. 50
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Q23. The sequence u₁, u₂, u₃, ... satisfies u₁ = 4 and u_n+1 = 2u_n − n for every integer n ≥ 1. Find u₁₀.
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Answer
Answer: A. 1035
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Q24. A sequence is defined by u₁ = 5 and u_n+1 = 2n − u_n for every positive integer n. What is u₁ + u₂ + … + u₂₀?
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Answer
Answer: C. 200
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Q25. The sequence u₁, u₂, u₃, ... is defined by u₁ = 4 and u_n+1 = u_n + n(n + 1) for every integer n ≥ 1. Find u₁₂.
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Answer: B. 576
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Q26. The sequence v₁, v₂, v₃, ... is defined by v₁ = 3 and v_n+1 = 5 − 2v_n for every integer n ≥ 1. Find v₆.
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Answer
Answer: C. −41
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Q27. A sequence has u₁ = 20 and u_n+1 = u_n + 2n − 9 for every integer n ≥ 1. What is the smallest term of the sequence?
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Answer: D. 4
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Q28. A sequence has v₁ = 3 and v_n+1 = v_n² − 2v_n + 2 for every integer n ≥ 1. What is v₆?
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Answer: D. 2³² + 1
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Q29. A sequence is defined by u₁ = 4 and u_n+1 = u_n + 3n² for every integer n ≥ 1. What is u₁₂?
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Answer: B. 1522
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Q30. A sequence is defined by v₁ = 6 and v_n+1 = n v_n/(n + 2) for every integer n ≥ 1. What is v₈?
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Answer: D. 1/6
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Q31. A sequence is defined by u₁ = 3 and u_n+1 = u_n + 3n for every integer n ≥ 1. Find u₄₀.
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Answer
Answer: C. 2343
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Q32. A sequence is defined by u₁ = 2, u₂ = 1 and u_n+2 = u_n+1 + 2u_n for every integer n ≥ 1. Find u₁₀.
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Answer
Answer: C. 511
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Q33. A sequence begins with t₁ = 2 and obeys t_n+1 = t_n + 4n − 1 for every integer n ≥ 1. Find t₂₀.
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Answer: A. 743
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Q34. The terms of a sequence are positive. They satisfy t₁ = 4 and t_n+1 = t_n + 2√(t_n) + 1 for every integer n ≥ 1. What is t₁₀?
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Answer: E. 121
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Q35. The terms of a sequence are given by t_n = an² + bn + c, where a, b and c are constants. Given that t₁ = 2, t₂ = 7 and t₄ = 29, find t₁₀.
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Answer: A. 191
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Q36. The sequence u₁, u₂, u₃, ... is defined by u₁ = 2 and u_n+1 = u_n/(1 + u_n) for n ≥ 1. What is u₆?
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Answer
Answer: E. 2/11
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Q37. In the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ..., each positive integer k appears exactly k times, in increasing order. What is the sum of the first 50 terms?
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Answer: C. 335
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Q38. A function g is defined on the positive integers by g(1) = 1 and, for every positive integer m, g(2m) = 3g(m) and g(2m + 1) = 3g(m) + 1. What is g(23)?
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Answer: D. 94
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Q39. The terms of a sequence satisfy u_n+1 − 2u_n + u_n−1 = 2 for every integer n ≥ 2, with u₁ = 3 and u₂ = 4. What is u₄₀?
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Answer
Answer: E. 1524
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Q40. The sequence u_n is defined by u₁ = 2 and u_n+1 = (1 + u_n)/(1 − u_n) for n ≥ 1. What is the product u₁ × u₂ × … × u₉₉?
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Answer
Answer: C. 3
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Q41. F(1) = 2. For n ≥ 2, F(n) = F(n − 1) + 3 if 2 divides n but 3 does not, F(n) = 2 F(n − 1) if 3 divides n but 2 does not, F(n) = F(n − 1) − 1 if 6 divides n, and F(n) = F(n − 1) + 1 otherwise. What is F(10)?
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Answer
Answer: B. 37
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Q42. The sequence t_n satisfies t₁ = 100 and t_n+1 = 100 √(t_n). It is known that t₆ = 10^a. What is a?
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Answer: B. 63/16
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Q43. A sequence satisfies t_n+1 = 12 t_n / t_n−1 for every n ≥ 2, with t₁ = 2 and t₂ = 6. What is t₂₀₂₄?
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Answer
Answer: E. 6
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Q44. The function f is defined on the positive integers by f(1) = 4, and for n ≥ 1, f(n + 1) = f(n)/2 when f(n) is even, while f(n + 1) = 3f(n) + 1 when f(n) is odd. What is f(1) + f(2) + … + f(100)?
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Answer
Answer: E. 235
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In the expansion of (1 + x)n, where n is a positive integer, the coefficients of xk, xk+1 and xk+2 are in the ratio 1 : 2 : 3. What is n?

What this topic tests

How it is assessed

Key methods

Arithmetic and geometric

A recurrence is not yet a formula

Common mistakes

Using the partial sum as the term

An infinite sum that does not converge

Worked example

The 5th term of an arithmetic sequence is 18 and the 11th term is 48. Find the 16th term.

Paper 1 questions

Q1. In the expansion of (1 + x)n, where n is a positive integer, the coefficients of xk, xk+1 and xk+2 are in the ratio 1 : 2 : 3. What is n?

Q2. The 5th term of an arithmetic sequence is 18 and the 11th term is 48. Find the 16th term.

Q3. Evaluate the sum from k = 1 to 7 of (3k + 2).

Q4. A geometric sequence has third term 54 and sixth term −2. What is the first term?

Q5. The sum of the first n terms of an arithmetic sequence is Sn = 5n2 − 2n. What is the common difference?

Q6. Find the coefficient of x2 in the expansion of (3 − 2x)5.

Q7. The nth term of a sequence is 1/[n(n + 2)]. Find the sum of its first 10 terms.

Q8. The sequence bn satisfies bn = 5 bn−1 − 6 bn−2 for every integer n ≥ 3, with b1 = 1 and b2 = 4. Which expression equals bn for every positive integer n?

Q9. For each positive integer n, let Tn = 1 × 1 + 2 × 3 + 3 × 5 + … + n(2n − 1). Which expression is equal to Tn?

Q10. A sequence of points is defined by P1 = (3, 1) and by the rule that Pn+1 is the image of Pn after a rotation of 90 degrees anticlockwise about the origin. What is P7?

Q11. A sequence is defined by a1 = 1 and an+1 = 2 an + 3. What is a6?

Q12. An arithmetic sequence has first term 5 and 40th term 83. What is the sum of its first 40 terms?

Q13. A geometric sequence has positive terms. Its 2nd term is 3 and its 6th term is 48. What is the common ratio?

Q14. What is the coefficient of x4 in the expansion of (x3 + 2x + 5)3?

Q16. Evaluate the sum from n = 1 to n = 23 of (−1)n+2 + (−1)n+4 + (−1)n+6 − (−1)n−1.

Q17. Two geometric series each have first term 6, and both converge. Adding corresponding terms produces a series S whose first three terms are 12, 5 and 13/6. Find the sum to infinity of S.

Q18. The first three terms of an arithmetic progression are 2p, q and p2, where p < 0. The first three terms of a geometric progression are 2p, p2 and q. Find the sum of the first 8 terms of the arithmetic progression.

Q19. The sequence xn is defined by x1 = 3 and xn+1 = (xn + 3)/(1 − xn). The first three terms are 3, −3 and 0. What is the value of x99?

Q20. In the expansion of (a + bx)4, the coefficient of x3 is 9 times the coefficient of x. Given that a and b are positive integers, what is the smallest possible value of a + b?

Q21. For each positive integer n, F(n) is (1/n) times the integral from 0 to n of (3n − 3x) dx, and G(n) is the sum of F(1) + F(2) + ... + F(n). What is the smallest positive integer n such that G(n) > 90?

Q22. In an arithmetic progression, the sum of the first 10 terms is 50 and the sum of the next 10 terms (terms 11 to 20) is 250. What is the sum of terms 21 to 30?

Q23. The non-zero constant k is chosen so that the coefficients of x4 in the expansions of (1 + kx2)5 and (k + x)7 are equal. What is k?

Q24. A sequence is defined by u1 = 2 and un+1 = 3un + 1 for n ≥ 1. Which expression gives un?

Q25. Find the sum to infinity of the series 1 + 3/2 + 5/4 + 7/8 + 9/16 + …, whose numerators are the odd numbers 1, 3, 5, … and whose denominators are the powers of 2.

Q26. The sequence is given by x1 = 5 and xn+1 = √(xn) for n ≥ 1. What is x40?

Q27. A sequence is defined by u1 = 3 and un+1 = 2un − 1. What is the sum of its first six terms?

Q28. The 1st, 2nd and 3rd terms of a geometric progression are also the 1st, 3rd and 4th terms, respectively, of an arithmetic progression. The sum to infinity of the geometric progression is 18. What is its first term?

Q29. Evaluate the sum from k = 1 to k = 20 of 1/[k(k + 1)].

Q30. A sequence satisfies xn+1 = (xn + p)/(xn + q), where p and q are real constants. Given x1 = 2, x2 = 5 and x3 = 8, find x4.

Q31. A geometric sequence has positive integer first term and integer common ratio r > 1. Its partial sums satisfy S12 = k S4 for a positive integer k. What is the smallest possible k?

Q32. Evaluate the sum from n = 0 to infinity of sin(nπ + π/6) / 3ⁿ.

Q33. How many of the nine coefficients in the expansion of (1 + 2x)⁸ are multiples of 32?

Q34. The integer k is chosen at random from the 20 non-zero integers from −10 to 10 inclusive. The geometric series S = 1 + 2/k + 4/k² + 8/k³ + … has common ratio 2/k. What is the probability that S is a finite number less than 3/2?

Q35. Find the value of 1² − 2² + 3² − 4² + ... + 99² − 100².

Q36. What is the coefficient of x2 in (3 − x2)[(1 + x + 2x2)4 − (1 + 5x3)3]?

Q37. The expansion of (2 + ax)4 is 16 + px + 216x² + qx³ + rx⁴, where a, p, q and r are positive real numbers. Find p + q + r.

Q38. What is the value of 1 × 2 + 2 × 2² + 3 × 2³ + … + 10 × 210?

Q39. The inequality (n + 3) + (2n + 6) + (3n + 9) + … + (20n + 60) > k holds for every integer n ≥ 0. Which statement must be true?

Q40. A geometric series has first term 6 and common ratio r, with −1 < r < 1. The sum to infinity of the cubes of its terms is 3/7 of the cube of its own sum to infinity. Find r.

Q41. A geometric sequence has first term 3 and common ratio 2. What is the 5th term?

Q42. The coefficient of x3 in the expansion of (2 + x + x2)5 is twice the coefficient of x2 in the expansion of (2 + ax)5. Find every possible value of the constant a.

Q43. The coefficient of x3 in the expansion of (3 + cx)5 is three times the coefficient of x2 in the expansion of (1 + cx)6. Given that c is nonzero, find c.

Q44. A geometric sequence with real common ratio has second term 6 and fifth term −48. What is the sum of its first six terms?

Q45. When (1 + x − x²)(1 + 3x)⁴ is expanded, what is the coefficient of x²?

Q46. A geometric sequence has real common ratio. Its second term is 4, and the sum of its second, third and fourth terms is 52. What is the sum of all the possible values of its first term?

Q47. A geometric series has first term 6 and sum to infinity 10. Find the common ratio.

Q48. The first three terms of an arithmetic sequence are k − 1, 2k + 1 and 4k − 1, where k is a constant. What is the sum of the first 10 terms of the sequence?

Q49. When (2 + x − x²)(1 − 3x)³ is expanded, what is the coefficient of x²?

Q50. A geometric sequence has second term 8 and fifth term −27. Find the first term.

Q51. A geometric series has first term 6 and sum to infinity 4. Find the common ratio.

Q52. The arithmetic series 69 + 65 + 61 + ... is added up one term at a time. What is the greatest value reached by the running total?

Q53. When (3 − x + 2x²)(2 + x)³ is expanded in powers of x, what is the coefficient of x²?

Q54. A geometric series has sum to infinity 27, and the sum of its first three terms is 26. Find the first term.

Q55. A geometric series has first term 4 and sum to infinity 5. Find the common ratio.

Q56. What is the sum of all the integers between 1 and 200 that leave remainder 2 when divided by 7?

Q57. Find the coefficient of x² in the expansion of (3 + x)(2 − x)⁴.

Q58. A geometric sequence has second term 3 and fifth term 192. Find its first term.

Q59. An infinite geometric series has first term 9 and sum 45. Find the common ratio.

Q60. Find the sum of all the odd integers between 100 and 200.

Q61. In the product (2 + 3x + x²)(1 + x)⁵, what is the coefficient of x³?

Q62. For every positive integer n, the sum of the first n terms of a sequence is 3n+1 + k, where k is a constant. The sequence is geometric. What is k?

Q63. The sequence has nth term un = 1/((2n − 1)(2n + 1)). Find the sum of its first 10 terms.

Q64. An arithmetic sequence has 3rd term 11 and 10th term 39. What is the sum of its first 20 terms?

Q65. When (2 − x + 3x²)(1 + x)5 is expanded, what is the coefficient of x²?

Q66. A geometric sequence has third term 48 and sixth term 6. Find its first term.

Q67. An infinite geometric series has first term 15 and sums to 9. Find the common ratio.

Q68. What is the sum of all positive integers less than 100 that are multiples of 4 or of 6?

Q69. When (1 + 2x − 3x²)(1 + 4x)³ is expanded, what is the coefficient of x²?

In the expansion of (1 + x)ⁿ, where n is a positive integer, the coefficients of x^k, x^k+1 and x^k+2 are in the ratio 1 : 2 : 3. What is n?

Q1. In the expansion of (1 + x)ⁿ, where n is a positive integer, the coefficients of x^k, x^k+1 and x^k+2 are in the ratio 1 : 2 : 3. What is n?

Q5. The sum of the first n terms of an arithmetic sequence is S_n = 5n² − 2n. What is the common difference?

Q6. Find the coefficient of x² in the expansion of (3 − 2x)⁵.

Q8. The sequence b_n satisfies b_n = 5 b_n−1 − 6 b_n−2 for every integer n ≥ 3, with b₁ = 1 and b₂ = 4. Which expression equals b_n for every positive integer n?

Q9. For each positive integer n, let T_n = 1 × 1 + 2 × 3 + 3 × 5 + … + n(2n − 1). Which expression is equal to T_n?

Q10. A sequence of points is defined by P₁ = (3, 1) and by the rule that P_n+1 is the image of P_n after a rotation of 90 degrees anticlockwise about the origin. What is P₇?

Q11. A sequence is defined by a₁ = 1 and a_n+1 = 2 a_n + 3. What is a₆?

Q14. What is the coefficient of x⁴ in the expansion of (x³ + 2x + 5)³?

Q16. Evaluate the sum from n = 1 to n = 23 of (−1)ⁿ⁺² + (−1)ⁿ⁺⁴ + (−1)ⁿ⁺⁶ − (−1)ⁿ⁻¹.

Q18. The first three terms of an arithmetic progression are 2p, q and p², where p < 0. The first three terms of a geometric progression are 2p, p² and q. Find the sum of the first 8 terms of the arithmetic progression.

Q19. The sequence x_n is defined by x₁ = 3 and x_n+1 = (x_n + 3)/(1 − x_n). The first three terms are 3, −3 and 0. What is the value of x₉₉?

Q20. In the expansion of (a + bx)⁴, the coefficient of x³ is 9 times the coefficient of x. Given that a and b are positive integers, what is the smallest possible value of a + b?

Q23. The non-zero constant k is chosen so that the coefficients of x⁴ in the expansions of (1 + kx²)⁵ and (k + x)⁷ are equal. What is k?

Q24. A sequence is defined by u₁ = 2 and u_n+1 = 3u_n + 1 for n ≥ 1. Which expression gives u_n?

Q26. The sequence is given by x₁ = 5 and x_n+1 = √(x_n) for n ≥ 1. What is x₄₀?

Q27. A sequence is defined by u₁ = 3 and u_n+1 = 2u_n − 1. What is the sum of its first six terms?

Q30. A sequence satisfies x_n+1 = (x_n + p)/(x_n + q), where p and q are real constants. Given x₁ = 2, x₂ = 5 and x₃ = 8, find x₄.

Q31. A geometric sequence has positive integer first term and integer common ratio r > 1. Its partial sums satisfy S₁₂ = k S₄ for a positive integer k. What is the smallest possible k?

Q36. What is the coefficient of x² in (3 − x²)[(1 + x + 2x²)⁴ − (1 + 5x³)³]?

Q37. The expansion of (2 + ax)⁴ is 16 + px + 216x² + qx³ + rx⁴, where a, p, q and r are positive real numbers. Find p + q + r.

Q38. What is the value of 1 × 2 + 2 × 2² + 3 × 2³ + … + 10 × 2¹⁰?

Q42. The coefficient of x³ in the expansion of (2 + x + x²)⁵ is twice the coefficient of x² in the expansion of (2 + ax)⁵. Find every possible value of the constant a.

Q43. The coefficient of x³ in the expansion of (3 + cx)⁵ is three times the coefficient of x² in the expansion of (1 + cx)⁶. Given that c is nonzero, find c.

Q62. For every positive integer n, the sum of the first n terms of a sequence is 3ⁿ⁺¹ + k, where k is a constant. The sequence is geometric. What is k?

Q63. The sequence has nth term u_n = 1/((2n − 1)(2n + 1)). Find the sum of its first 10 terms.

Q65. When (2 − x + 3x²)(1 + x)⁵ is expanded, what is the coefficient of x²?

Q72. The nth term of a sequence is 2ⁿ + 3n. What is the sum of the first 10 terms of the sequence?

Q73. A sequence is given by a_n = (n + 2)/(n + 5). What is the product of the first 30 terms, starting at n = 1?

Q74. The sum of the first n terms of a sequence is S_n = 3n² − n. What is the 10th term?

Q76. A sequence is defined by u₀ = 1 and, for every integer n ≥ 1, u_n equals twice the sum of u₀ through u_n−1. Find the sum from r = 0 to infinity of 1/u_r.

Q77. Find the coefficient of x⁶ in the expansion of (1 + x + x²)⁷.

Q78. How many solutions does the equation cos(2x) + cos²(2x) + cos³(2x) + ... = 1 have in the interval 0 ≤ x < 2π?

Q79. What is the coefficient of x² in the expansion of (1 + x)(1 + 2x)(1 + 3x) ... (1 + 10x)?

Q83. Let floor(t) denote the greatest integer less than or equal to t. The sequence is defined by a_n = floor(√n). What is the sum of the first 48 terms, from n = 1 to n = 48?

Q84. A positive number k is chosen so that k + 2, k² + k and k² + 5k + 3 are the first three terms of an arithmetic sequence. What is the 11th term of that sequence?

Q87. An arithmetic sequence satisfies u₃ + u₈ + u₁₃ = 51 and u₅ × u₁₁ = 280. What is the greatest possible value of u₁?

Q89. A geometric sequence of real numbers satisfies u₂ × u₉ = 8. What is the product of the first 10 terms, u₁ × u₂ × ... × u₁₀?

Q90. A sequence is defined by b₀ = 2 and, for every integer n ≥ 1, b_n = b₀ × b₁ × ... × b_n−1 + 1. Find the sum of the reciprocals 1/b_r from r = 0 to infinity.

Q91. A sequence satisfies y₁ = 1, y₂ = 4 and y_n+1 = (1 + y_n) / y_n−1 for every integer n ≥ 2. What is y₂₀₂₅?

Q93. For each integer n ≥ 1, let u_n = (−1)ⁿ⁺¹ × 3n, and let w_n be the sum of u₁ through u_n. Find the value of n for which w_n = 750.

Q94. A sequence is defined by a₁ = x and a_n+1 = −1 − 1/(2a_n) for every integer n ≥ 1, where x is chosen so that every term is defined. What is the smallest positive integer p such that, for every such choice of x, a_n+p = a_n for every n?

Q95. A sequence is defined by a₁ = 1 and a_n+1 = (a_n + k)/(1 − k a_n) for every integer n ≥ 1, where k is a nonzero real constant chosen so that every term is defined. For which values of k does the sequence have minimal period 3?

Q96. What is the coefficient of x² in the expansion of (2 + x)[(1 − 2x + 3x²)⁴ − (1 + x³)⁵]?

Q97. What is the coefficient of x² in the expansion of (1 − x + 4x²)⁶?

Q98. In the expansion of (2 + x)⁷, what is the sum of the coefficients of the even powers of x, including the constant term?

Q99. What is the coefficient of x⁴ in the expansion of (1 − x)³ (1 + x)⁷?

Q100. A sequence satisfies u_n = 3u_n−1 + 2u_n−2 for every integer n ≥ 3, with u₁ = 1 and u₂ = 5. What is the limit of u_k / u_k−1 as k tends to infinity?

Q1. The constant term in the expansion of (3x + a/x)⁴ is 216, and a is positive. What is a?

Q2. The sequences a_k and b_k are arithmetic, and a_k/b_k has the same value for k = 1, 2, 3 and 4. Given a₁ = 180, a₄ = 72 and b₁ = 90, what is b₂?

Q3. A sequence is defined by a₁ = 2 and a_n+1 = a_n/(1 + a_n) for n ≥ 1. What is a₅₀?

Q4. The sequence a_n is geometric with real terms, a₃ = −2 and a₇ = −162. What is a₅?

Q7. A sequence u₀, u₁, u₂, ... is defined by u₀ = 1 and, for n ≥ 1, u_n equals the integral from 0 to 1 of 6x u_n−1 dx. What is the value of u₁₂?

Q13. Find the coefficient of x⁵ in (2 + x)⁵ (1 + x + x² + x³ + x⁴ + x⁵).

Q15. A sequence is defined by u₁ = a, u₂ = b and u_n+2 = u_n+1 − u_n for n ≥ 1, where a and b are positive integers. Which of these statements must be true? I: u₂₀ = b. II: u₁ + u₂ + u₃ + u₄ + u₅ + u₆ = 0. III: u₂₀₃ is positive.

Q16. The sequence a_n satisfies a₁ = 3 and a_n + a_n+1 = 4n for every integer n ≥ 1. What is a₁ + a₂ + a₃ + ... + a₄₁?

Q20. A sequence has t₁ = 2 and t₂ = 3, and for n ≥ 2, t_n+1 = (1 + t_n)/t_n−1. What is t₁ + t₂ + ... + t₂₀₂₄?

Q22. The function f is defined on the positive integers by f(1) = −1 and, for every integer m ≥ 1, f(2m) = (f(m))^{m + 1} and f(2m + 1) = (f(m))^m. Find f(1) + f(2) + ... + f(60).

Q23. The sequence u₁, u₂, u₃, ... satisfies u₁ = 4 and u_n+1 = 2u_n − n for every integer n ≥ 1. Find u₁₀.

Q24. A sequence is defined by u₁ = 5 and u_n+1 = 2n − u_n for every positive integer n. What is u₁ + u₂ + … + u₂₀?

Q25. The sequence u₁, u₂, u₃, ... is defined by u₁ = 4 and u_n+1 = u_n + n(n + 1) for every integer n ≥ 1. Find u₁₂.

Q26. The sequence v₁, v₂, v₃, ... is defined by v₁ = 3 and v_n+1 = 5 − 2v_n for every integer n ≥ 1. Find v₆.

Q27. A sequence has u₁ = 20 and u_n+1 = u_n + 2n − 9 for every integer n ≥ 1. What is the smallest term of the sequence?

Q28. A sequence has v₁ = 3 and v_n+1 = v_n² − 2v_n + 2 for every integer n ≥ 1. What is v₆?

Q29. A sequence is defined by u₁ = 4 and u_n+1 = u_n + 3n² for every integer n ≥ 1. What is u₁₂?

Q30. A sequence is defined by v₁ = 6 and v_n+1 = n v_n/(n + 2) for every integer n ≥ 1. What is v₈?

Q31. A sequence is defined by u₁ = 3 and u_n+1 = u_n + 3n for every integer n ≥ 1. Find u₄₀.

Q32. A sequence is defined by u₁ = 2, u₂ = 1 and u_n+2 = u_n+1 + 2u_n for every integer n ≥ 1. Find u₁₀.

Q33. A sequence begins with t₁ = 2 and obeys t_n+1 = t_n + 4n − 1 for every integer n ≥ 1. Find t₂₀.