A geometric progression has first term a and common ratio r, where r > 0. The second term of this progression is 8. The sum of the third and fourth terms is 160. Show that r satisfies the equation r² +r-20 = 0.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B2] B1 for each ar = 8 and ar² + ar³ = 160 soi
2. [M1] Correct unsimplified quadratic equation in r e.g. [a = 8/r and so] 8/r * r² + 8/r * r³ = 160 oe
3. [A1] Correct simplification to r² + r −20 = 0
4. [M1] Factorises or solves: (r + 5)(r – 4) = 0
5. [A1] r = 4
6. [A1] a = 2
7. [B1] p + 2(q - 1) = 14 oe
8. [B1] q/2 {2p+4(q-1)} = 168 oe
9. [M1] Eliminates p or q e.g. q/2{2(14-2(q-1))+4(q-1)} = 168 OR simplifies Sq: q{p+2(q-1)} =168 and writes q(14)=168 condone one error in rearrangement of either equation before substituting
10. [M1] Correctly solves their equation for their p or their q dep on previous M1
11. [A2] A1 for q = 12 or p = −8 q = 12 and p = -8