A geometric series has first term a and common ratio r. The second term of the series is 12 and the sum to infinity is 64.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. Part (a): B1 for writing down ar = 12 (or equivalent).
2. Part (a): B1 for writing down a / (1 - r) = 64 (or equivalent).
3. Part (a): M1 for eliminating 'a' from their two equations (e.g., substitute a = 12/r into the second equation).
4. Part (a): A1 for correctly reaching 12 / (r(1 - r)) = 64 leading to 12 = 64r - 64r² and then 3r² + 16r - 16 = 0.
5. Part (b): M1 for solving the quadratic equation 3r² + 16r - 16 = 0 (e.g., using formula or factorization).
6. Part (b): A1 for r = 2/3 (and rejecting r = -8/3 as |r| < 1 is required for sum to infinity).
7. Part (b): M1 for substituting their valid r into ar = 12 to find a.
8. Part (b): A1 for a = 18.