The first three terms of a geometric progression are a, b and c respectively, where a, b and c are positive constants. The first three terms of an arithmetic progression are a, b and -3c respectively.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.

1. [B1] States condition for AP: 2b = a + (-3c) or equivalent. OR condition for GP: b² = ac or equivalent.
2. [M1] Forms a single equation in a and c only by eliminating b from the AP and GP conditions.
3. [A1] a² - 10ac + 9c² = 0 (AG - convincing proof required, e.g., showing (a-3c)² = 4ac leads to the result).
4. [M1] Substitutes a=9 into a² - 10ac + 9c² = 0 and solves for c (9c² - 90c + 81 = 0 => c² - 10c + 9 = 0).
5. [B1] Identifies c=1 as the smaller possible value (reject c=9).
6. [M1] Finds the common ratio r using b²=ac and 2b=a-3c, and a=9, c=1 (e.g., b²=9, 2b=9-3=6 => b=3, so r=b/a=3/9=1/3).
7. [DM1] Uses sum to infinity formula S_inf = a/(1-r) with a=9 and their |r|<1.
8. [A1] S_infinity = 27/2 or 13.5 or 13 1/2.
9. [B1] Finds common difference d for AP. (a=9, b=3, -3c=-3. d = b-a = 3-9 = -6).
10. [M1] Uses arithmetic series sum formula S_n = n/2 (2a + (n-1)d) with n=20, a=9, and their d.
11. [A1] S_20 = -960.