The sum, Sₙ, of the first n terms of an arithmetic progression is given by Sₙ = 32n – n². Find the first term and the common difference.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.

1. [1] Set n = 1 to find S₁ = 32(1) – 1² = 31. This is the first term, a. co
2. [1] Set n = 2 to find S₂ = 32(2) – 2² = 60. The second term is S₂ - S₁ = 60 - 31 = 29. Correct method
3. [1] The common difference d = (second term - first term) = 29 - 31 = -2. co
4. [1] Form an equation for sum to infinity: a/(1-r) = 20. co
5. [1] Form an equation for the sum of the first two terms: a + ar = 12.8. co
6. [1] Eliminate one variable (a or r) to form an equation in the other. 'Correct' elimination to form equation in a or r
7. [2] Solve for r and a. r = 0.6 and a = 8. Complete method leading to a. Condone a = 8 and 32 (if r=-0.6 solution is kept)