Given that $k$ is a real constant, find $\displaystyle \sum_{r=5}^{30}[2^{1-2r}-(k-2r)]$, l

Question

Given that $k$ is a real constant, find $\displaystyle \sum_{r=5}^{30}[2^{1-2r}-(k-2r)]$, leaving your answer in the form $A(1-B)+Ck+910$, where $A$, $B$ and $C$ are exact real constants.[3] Another series is such that $$\sum_{r=1}^{N}\frac1{r(r+1)(r+2)}=\frac12\left(\frac12-\frac1{(N+1)(N+2)}\right).$$ Find $\displaystyle \sum_{r=1}^{\infty}\frac1{r(r+1)(r+2)}$. Explain your answer clearly.[1] Find $\displaystyle \sum_{r=13}^{N}\frac1{3r(r-1)(r-2)}$ in terms of $N$.[3] A sequence of positive numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation $$x_{n+1}=\sqrt{10-3x_n},\qquad n=1,2,3,\ldots$$ Given that $x_4=1$, find the exact value of $x_1$.[2] It is also known that the sequence converges to $l$. Find the value of $l$.[2] Prove that $(x_{n+1})^2-l^2=3(l-x_n)$.[1] Use the result in part (iii) to show that if $x_n<l$, then $x_{n+1}>l $. [2]

Accepted Answer

(a) $\frac1{384}(1-4^{-26})-26k+910$ (b)(i) $\frac14$ (b)(ii) $\frac1{792}-\frac1{6N(N-1)}$ (c)(i) $x_1=\frac{89}{27}$ (c)(ii) $l=2$ (c)(iii) identity proven (c)(iv) $x_nl$

2026 ASRJC P1 Q10

2026 ASRJC P1 Q10

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