A geometric progression has a 4th term of 8k⁶/27 and a 6th term of 32k¹⁰/243, where k is a constant. The common ratio of this geometric progression is positive.

Mark scheme

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

1. [B1] ar³ = 8k⁶/27, ar⁵ = 32k¹⁰/243 soi (May be implied by a correct 5th term 16k⁸/81)
2. [M1] Divide ar⁵ by ar³ to find r² (for solution of their equations to obtain either r=mk² or a=c, where m is an unsimplified numeric constant and c is a non-zero constant, e.g. Use of r² = 6th term / 4th term)
3. [A1] r = 2k²/3
4. [A1] a=1
5. [B1] 1 / (1 - 2k²/3) = 3 soi (For use of sum to infinity formula with their values of a where a is numeric and r which is in terms of k²)
6. [B1] k = ±1 (B1 for each must be from correct work)