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TMUA sequences and series questions
All 35 sequences and series questions from the TMUA past papers, 2016 to 2023, newest first, each with a worked solution.
2023
- P1 Q4 · Evaluate ∑_n=0^∞ (sin(nπ + (π)/(3)))/(2ⁿ)
- P1 Q6 · In the simplified expansion of (2 + 3x)¹², how many of the terms have a coefficient that is… hard
- P1 Q17 · A circle C_n is defined by x² + y² = 2n(x + y) where n is a positive integer. C₁ and C₂ are drawn… hard
- P1 Q18 · You are given that S = 4 + (8k)/(7) + (16k²)/(49) + (32k³)/(343) + ⋯ + 4((2k)/(7))ⁿ + ⋯ The value… hard
2022
- P1 Q5 · The terms x_n of a sequence follow the rule x_n+1 = (x_n + p)/(x_n + q) where p and q are real…
- P1 Q8 · A geometric sequence has first term a and common ratio r, where a and r are positive integers and…
- P1 Q11 · Evaluate ∑_n=1¹⁰⁰log₁₀(3^,1-n)
- P2 Q2 · Find the coefficient of the x⁵ term in the expansion of (1+x)⁵ × ∑_i=0⁵ x^i
- P2 Q8 · A selection, S, of n terms is taken from the arithmetic sequence 1, 4, 7, 10, …, 70. Consider the… hard
2021
- P1 Q3 · An arithmetic progression and a convergent geometric progression each have first term (1)/(2) The…
- P1 Q16 · Consider the expansion of (a + bx)ⁿ The third term, in ascending powers of x, is 105x² The fourth… hard
2020
- P1 Q4 · The 1 st , 2 nd and 3 rd terms of a geometric progression are also the 1 st , 4 th and 6 th terms,…
- P1 Q13 · Find the coefficient of x²y⁴ in the expansion of (1 + x + y²)⁷.
- P2 Q14 · An arithmetic sequence T has first term a and common difference d, where a and d are non-zero…
2019
- P1 Q3 · Find the coefficient of x in the expression (1+x)⁰ + (1+x)¹ + (1+x)² + (1+x)³ + ⋯ + (1+x)⁷⁹ +…
- P1 Q4 · The sequence x_n is given by x₁ = 10 x_n+1 = √(x_n) for n ≥ 1 What is the value of x₁₀₀? [Note…
- P1 Q5 · S is a geometric sequence. The sum of the first 6 terms of S is equal to 9 times the sum of the…
- P2 Q1 · Find the coefficient of the x⁴ term in the expansion of x²(2x + (1)/(x))⁶
- P2 Q11 · An arithmetic series has n terms, all of which are integers . The sum of the series is 20. Which…
2018
- P1 Q2 · An arithmetic progression has first term a and common difference d. The sum of the first 5 terms…
- P1 Q7 · The non-zero constant k is chosen so that the coefficients of x⁶ in the expansions of (1 + kx²)⁷…
- P1 Q8 · The sum to infinity of a geometric progression is 6. The sum to infinity of the squares of each…
- P2 Q2 · Find the value of the constant term in the expansion of (x⁶ - (1)/(x²))¹²
- P2 Q16 · In this question, x₁, x₂, x₃, … is an arithmetic progression , all of whose terms are integers.…
2017
- P1 Q7 · The first three terms of an arithmetic progression are p, q and p² respectively, where p The first…
- P1 Q11 · The sequence x_n is defined by the rules x₁ = 7 x_n+1 = (23x_n - 53)/(5x_n + 1) The first three…
- P1 Q13 · In the expansion of (a + bx)⁵ the coefficient of x⁴ is 8 times the coefficient of x². Given that a…
- P1 Q17 · The two functions F(n) and G(n) are defined as follows for positive integers n: F(n) = (1)/(n)∫₀ⁿ…
- P2 Q3 · The first term of a geometric progression is 23 and the fourth term is (9)/(4). What is the sum to…
- P2 Q15 · The function f is defined on the positive integers as follows: f(1) = 5, and for n ≥ 1, f(n+1) =…
2016
- P1 Q4 · The sequence a_n is defined by the rule a_n = (-1)ⁿ - (-1)ⁿ⁻¹ + (-1)ⁿ⁺² for n ≥ 1. Find the value…
- P1 Q14 · The terms of an infinite series S are formed by adding together the corresponding terms in two… hard
- P1 Q19 · The coefficient of x³ in the expansion of (1 + 2x + 3x²)⁶ is equal to twice the coefficient of x⁴… hard
- P2 Q6 · The sequence of functions f₁(x), f₂(x), f₃(x), … is defined as follows: f₁(x) = x¹⁰ f_n+1(x) = x…
- P2 Q12 · The first term of an arithmetic sequence is a and the common difference is d. The sum of the first…