The first, second and third terms of an arithmetic progression are 4k, k² and 8k respectively, where k is a non-zero constant.

Mark scheme

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

1. [1] M1 OE Forming an equation in k only, clearly using the first 3 terms of an AP. 8k-k² =k² −4k or 2k² =4k+8k or 4k+2(k² - 4k) = 8k
2. [1] A1 Condone extra 'solution' k = 0. Note: can be done by inspection or with no working. [2k(k-6)=0 =>] k=6 [or k=0]
3. [1] *M1 Using a correct method to find a and d from their k. SOI. May have been found in (a)(i) but must be used in (ii). d [=8k-k² or 2k or k² - 4k]=12 and a[=4k] = 24
4. [1] DM1 Use of correct sum formula with n = 20 and their a and d. S20 = 20/2 (2x24+19x12) or = 20/2 (24 + 252)
5. [1] A1 S20 = 2760
6. [1] *M1 Using a correct method to find r. Condone ±. ar³ = 36 and ar⁵ = 6 or r² = 1/6
7. [1] *M1 Using a correct method to find a. Condone ±. a = 36 / (1/√6)³ = 36x√6², 216√6 or 1296/√6
8. [1] DM1 Using the correct formula with their a and their |r|<1. S∞ = (216√6) / (1 - 1/√6)
9. [1] A1 OE in the required form. r=0.408 => a=529 => S∞ = 894 scores 4/5. S∞ = 1296 / (√6-1) or 7776 / (√216-6)