An arithmetic progression has twelve terms. The sum of the first three terms is -36 and the sum of the last three terms is 72. Find the first term and the common difference.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

1. [B3] B2 for one correct simplified equation or B1 for a+a+d+a+2d=-36 for Correct pair of simplified linear equations in a and d with terms collected, e.g., 3a+3d=-36 isw or a + d = -12 isw and 3a+30d=72 isw or a + 10d = 24 isw
2. [M1] FT their linear equations in a and d providing at least B1 earned and the equations have a solution for Solves two linear equations for d or a e.g. 27d=108 → d = ...
3. [A1] d = 4 and a = -16 nfww
4. [B3] where * is any inequality sign or =; B2 for [1](1.2^n-1) / (1.2-1) * 500 or B1 for r = 1.2 soi for 1.2ⁿ * 101
5. [M1] FT 1.2ⁿ * their 101 providing B2 has been awarded and (their 101) > 0 for nlog1.2*log101 or log1.2101 soi
6. [A1] n = 26