A telescoping increment · source reference

Printed question

A sequence $u_1,u_2,u_3,\ldots$ is such that $u_1=\frac32$ and

$$u_{n+1}=u_n+\frac1{(n+1)(n+2)},\qquad n\in\mathbb Z^+.$$

(i) Find the values of $u_2,u_3,u_4$. [2]

(ii) Form a conjecture for $u_n$ in terms of $n$. [1]

Printed working

(i) Given $u_1=\frac32$,

$$u_2=\frac53,\quad u_3=\frac74,\quad u_4=\frac95.$$

(ii) Conjecture: $u_n=\frac{2n+1}{n+1}$ for $n\in\mathbb Z^+$.

Printed answer key

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

Mathematical review and qualifications

(i) $u_2=3/2+1/6=5/3$, $u_3=5/3+1/12=7/4$, $u_4=7/4+1/20=9/5$.

(ii) The conjecture is $u_n=(2n+1)/(n+1)=2-1/(n+1)$. It is verified by $u_1=3/2$ and $u_{n+1}-u_n=1/(n+1)-1/(n+2)$, exactly the prescribed increment.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: RECURRENCE RELATIONS. [2011/DHS/Prelim/II/4b(i),(ii)] Question PDF pages 1; solution PDF pages 1. The original files remain private QA records outside this native content.