The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

1. [1] M1 For (a): Uses the correct sum of an arithmetic progression formula (S_n = n/2(2a + (n-1)d)) to find d from S₄ = 57 (e.g., 57 = 4/2(2(12) + 3d) → d = 1.5)
2. [1] A1 For (a): Correct common difference d = 1.5
3. [1] M1 For (a): Uses the correct n-th term of an arithmetic progression formula (T_n = a + (n-1)d) to find n from T_n = 48 (e.g., 48 = 12 + (n-1)1.5)
4. [1] A1 For (a): Correct number of terms n = 25
5. [1] B1 For (b): Sets up equation for the third term (ar² = 4a) and finds r = ±2
6. [1] B1 For (b): Sets up equation for sum of first six terms (S₆ = k * a), e.g., a(r⁶-1)/(r-1) = ka
7. [1] B1 For (b): Finds k = 63 (for r = 2)
8. [1] B1 For (b): Finds k = -21 (for r = -2)