Arithmetic & Geometric Progressions

Source metadata

Repeated running header: 2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS. The first question and solution pages also carry the bold topic heading ARITHMETIC & GEOMETRIC PROGRESSIONS. Solution pages are numbered 1–32. Question printed pages are 89–96. Repeated “Solution” and “Answers” headings are represented by the corresponding native section headings.

Question index

• 12-01: A variable geometric ratio
• 12-02: A relative geometric-tail error
• 12-03: Link an AP with a convergent GP
• 12-04: Odd-indexed terms and an alternating tail
• 12-05: Selected GP terms form an AP
• 12-06: Half the infinite sum and alternate indices
• 12-07: Eliminate two progression parameters
• 12-08: Alternating terms, even terms and products
• 12-09: Odd-sized groups of geometric terms
• 12-10: A half-tail identity and triangular rows
• 12-11: A filling machine and an insufficient tail
• 12-12: Positive ribbon lengths and a bounded GP
• 12-13: AP lap times against GP lap times
• 12-14: Remove multiples and sum a bouncing path
• 12-15: Declining sales prices and holding costs
• 12-16: Selected terms and a rabbit–fox chase
• 12-17: Monthly deposits and the first crossing day
• 12-18: A reducing loan and a final smaller payment
• 12-19: A declining income against rising costs
• 12-20: Out-and-back circuits and changing growth rates

Printed answer index

1. $\frac{3x-2}{12x(1-x)}$

2. 20

3. (b) $\frac72a$; (c) 25; (d) least $k=12$

4. (a)(iii) $u_1=e^2-3,u_3=5e^2-3,u_5=9e^2-3$; 7. (b)(i) $x<0$; (ii) −0.00181 (3 s.f.)

5. (i) $r=0.90824$ (5 s.f.) =0.908 (3 s.f.); (ii) 21.8 (3 s.f.); (iii) −33.2

6. (a) $\frac14$; (b) $\frac{25}{12}$; $-2\ln5$

7. (i) $r=0.3435$ (4 d.p.)

8. (ii) $-\ln4$; (iii) 56

9. $T_n=\frac1a(a-1)^{n-1}(2-a)$, $n\in\mathbb Z^+$; (i) $n^2$; (ii) $\frac1a(a-1)^{110}(2-a)$; (iii) $0<a<2$; 12

10. (b)(ii) $\frac{n(n-1)(n^2-n+2)}4$

11. (i) $21\mathrm{m}^3$; (ii)(a) $838\mathrm{m}^3$ (3 s.f.); (b) No

12. (a) 20; (b) 0.84; 17; (c) 2633cm (nearest cm)

13. (ii) $1500(1.04^n-1)$; (iii) 25 laps; (iv) marathon runner; (v) 4th lap

14. (a) 1352737; (b)(i) $a^nH$; (ii) 11; (iii) $7H$

15. (i) $500/k$; (iii) $125(500+k)/k$; (iv) 25

16. (a)(i) $1/5$; (ii) $-625a/72$; (b)(i) 54; (ii) 150 leaps; 95m

17. (a) 862.50; (b)(i) $k(1.002^{24}-1)$; (ii) 903; (iii) 1 September 2023

18. (ii) 31 May 2043; 407.70 dollars (2 d.p.); (iii) 2220.42 dollars (2 d.p.)

19. (iii) $5000(1-0.9^n)+600(0.9)^{n-1}$; (iv) 2018

20. (i) 80; (ii) $50(3^n-1)$; (iii) running away from O at distance 5600m; (iv) $r\ge5.22$ (3 s.f.)

Calculator display conventions

Repeated graph/table status strips show NORMAL FLOAT AUTO REAL RADIAN MP (as applicable), PRESS ↑ FOR ΔTbl, Plot1 Plot2 Plot3, Y1/Y2, and Intersection or Zero. These calculator display labels and the associated mathematical equations, root/intersection readouts, and table values are expressed in each question’s native sections. The controls explain the original calculator workflow; they are not interactive screenshots.