An arithmetic progression has first term t and common difference 1.5. The 4th, 8th and 20th terms of this arithmetic progression form the 1st, 2nd and 3rd terms of a geometric progression.

Mark scheme

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

1. [B1 condone a for t] (a) At least two correct terms from: t+4.5 t+10.5 t+28.5 soi
2. [M1 FT their distinct terms of the form t + 1.5k where k is a positive integer] (a) r = (t+10.5)/(t+4.5) or r = (t+28.5)/(t+10.5) or r² = (t+28.5)/(t+4.5) oe
3. [A1 or (t+4.5) × (t+28.5)/(t+10.5) = t+10.5 oе] (a) (t+10.5)/(t+4.5) = (t+28.5)/(t+10.5) oe
4. [A1 must be seen] (a) Clears algebraic fractions and expands e.g. t² +21t+110.25=t² +33t +128.25
5. [A1 and no other values dep on all previous marks awarded] (a) -1.5 oe nfww
6. [B1] (a) Alternative method: At least two correct terms from: t+4.5 t+10.5 t+28.5 soi
7. [M1 FT their distinct terms of the form t + 1.5k where k is a positive integer] (a) One correct equation in a and r e.g. a -4.5 = ar – 10.5 or ar-10.5 = ar² – 28.5 or a – 4.5 = ar² – 28.5 oe
8. [A1 OR correct equation in a e.g. a+24=a(a+6/a)² ] (a) Correct equation in r e.g. 6/(r-1) = 18/(r(r-1)) OR 18/(r²-r) - 4.5 = 18r/(r²-r) - 10.5
9. [A1 must be seen OR a²+24a = a²+12a+36 and a = 3] (a) [6r=18] r = 3 OR 18-4.5(r² – r) =18r-10.5(r² – r) and e.g. t = 18r/(r²-r) -10.5 or a = 3
10. [A1 and no other values dep on all previous marks awarded] (a) -1.5 oe nfww
11. [M1] (b) for their(-1.5)+their10.5 / their(-1.5)+their4.5 oe or for their a + their10.5 – their4.5 / their a oe or for their 28.5-their10.5 / their(-1.5)+their10.5 +1 oe
12. [A1] (b) 3 nfww
13. [FT their t providing their t ≠ 0 or their a providing their a ≠ 0 or 1]