A geometric series has first term `a` and common ratio `r`. The second term of the series is `4` and the sum to infinity of the series is `18`. (a) Show that `9r^2 - 9r + 2 = 0`. (b) Find the two possible values of `r`. (c) Find the corresponding two possible values of `a`.

Mark scheme

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

1. B1 for `ar = 4`.
2. B1 for `a/(1-r) = 18`.
3. M1 for combining equations to eliminate `a` or `r`. e.g. `4/r / (1-r) = 18` or `a = 18(1-r)` and `18(1-r)r = 4`.
4. A1 for `9r^2 - 9r + 2 = 0` (given answer).
5. M1 for solving 3 term quadratic in `r` using formula or factorisation.
6. A1 for `r=1/3` and `r=2/3`.
7. M1 for substituting one of their `r` values into a correct equation for `a`.
8. A1 for `a=12` (for `r=1/3`) and `a=6` (for `r=2/3`).