Recurrence Relations

Source metadata

Repeated running header: 2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS. The first question and solution pages also carry the bold topic heading RECURRENCE RELATIONS. Solution pages are numbered 1–13. Question printed pages are 97–101. Repeated “Solution” and “Answers” headings are represented by the corresponding native section headings.

Question index

• 13-01: Recover terms from a cumulative sum
• 13-02: A telescoping increment
• 13-03: Alternation around a fixed point
• 13-04: Shift an affine recurrence into a GP
• 13-05: A rational recurrence with period four
• 13-06: Sum polynomial and exponential increments
• 13-07: A logarithmic fixed-point iteration
• 13-08: A square-root iteration crosses its limit
• 13-09: A rational map approaches two
• 13-10: Halve the error at every step
• 13-11: A conditional limit in a two-cycle map
• 13-12: An exponential denominator creates a stable limit
• 13-13: An alternating fifth-root iteration

Printed answer index

1. (i) $\frac{7+c-14n}2$; (ii) $U_n-7$

2. (i) $u_2=5/3,u_3=7/4,u_4=9/5$; (ii) $u_n=(2n+1)/(n+1)$, $n\in\mathbb Z^+$

3. (a)(i)(A) Initially decreases and subsequently alternates, converging to 2. (B) Constant value 2. (a)(ii) 9; (b)(i) $c+p=2$; (ii) 12

4. (i) $p=5/2,q=2$; (ii) $a_n=\frac43[(5/2)^{n-1}-1]$, $n\ge3$

5. (b)(i) $u_1=4,u_2=-1,u_3=2/3,u_4=3/2,u_5=4$

6. −1, 2; increasing and divergent

7. (i) $\alpha=1.052,\beta=5.505$

8. (i) 2

9. (i) 2; (iii) $u_n\to2$ as $n\to\infty$

10. (a)(i) 2; (iii) increases and converges to 2

11. (i) $l=\frac{-1+\sqrt7}2$

12. (i) $\alpha=\frac{\lg(4/3)}{\lg2}$; (iii) $L=\frac{\lg(4/3)}{\lg2}$

13. (i) 1.392 (3 d.p.); (iv) $0<x_2<\alpha<x_3<x_1$; oscillates and converges to 1.392

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.