Sequences and series carry about 25 of the 150 marks in Paper 1. First decide which type of pattern you have: check for a constant difference, a constant second difference or a constant ratio.
Question 1: Arithmetic sequence
For 5; 9; 13; …, find the general term and T₂₀. Here a = 5 and d = 4, so Tₙ = 5 + (n − 1)(4) = 4n + 1. Answer: Tₙ = 4n + 1 and T₂₀ = 81.
Question 2: Quadratic pattern
Find the general term of 3; 8; 15; 24; …. The first differences are 5; 7; 9 and the second difference is 2, so 2a = 2 and a = 1. From 3a + b = 5, b = 2. From a + b + c = 3, c = 0. Answer: Tₙ = n² + 2n.
Question 3: Geometric sequence
For 2; 6; 18; …, find T₆. Here a = 2 and r = 3, so T₆ = 2(3)⁵. Answer: 486.
Question 4: Arithmetic series
Find the sum 4 + 7 + 10 + … + 31. Tₙ = 4 + (n − 1)(3) = 31 gives n = 10. Sₙ = n/2 (a + l) = 10/2 (4 + 31). Answer: 175.
Question 5: Sum to infinity
Find the sum to infinity of 8 + 4 + 2 + …. Here r = ½, which lies between −1 and 1, so the series converges. S∞ = a / (1 − r) = 8 / ½. Answer: 16.
Question 6: Sigma notation
Evaluate Σ (2k + 1) from k = 1 to k = 4. Write out the terms: 3 + 5 + 7 + 9. Answer: 24.
Question 7: When a series converges
For which values of x does the series 1 + (x − 1) + (x − 1)² + … converge? Here r = x − 1, and a convergent series needs −1 < r < 1. Add 1 to each part of the inequality. Answer: 0 < x < 2.
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