A geometric progression has third term 4.5 and sixth term 15.1875. Find the first term and the common ratio.

Mark scheme

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

1. [B1] ar² = 4.5 and ar⁵ = 15.1875 soi
2. [M1] Correctly eliminates one unknown using correct equations e.g. (ar⁵)/(ar²) = 15.1875/4.5 or r³ = 15.1875/4.5 oe
3. [A2] r = 1.5, a = 2 (A1 for either)
4. [B2] S15 = (2(1-1.5^15))/(1-1.5) and S25 = (2(1-1.5^25))/(1-1.5) oe (M1 FT their a and r for S_n)
5. [M1] Correct plan: S25 - S15 oe attempted
6. [A1] 99253
7. [M1] first term = (their2)(their1.5)^15 and an attempt at S10 (Alternative 1)
8. [B2] Correct sum (875.7...(1-1.5^10))/(1-1.5) oe (M1 FT their first term and r for S10)
9. [A1] 99253 (Alternative 1)
10. [M3] Correct sum: 2(1.5)^15 + 2(1.5)^16 + ... + 2(1.5)^24 oe or 2(1.5)^15{1 + 1.5 + ... + (1.5)^9} (Alternative 2)
11. [A1] 99253 (Alternative 2)