The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by \[S_n=\f

Question

The sum, $S_n$, of the first $n$ terms of a sequence $u_1,u_2,u_3,\ldots$ is given by $$S_n=\frac16+\frac1{n+3}-\frac2{n+4}.$$ Give a reason why the series $\displaystyle \sum_{r=1}^{\infty}u_r$ converges and write down the value of the sum to infinity.[2] Show that $\displaystyle u_n=\frac{n}{(n+2)(n+3)(n+4)}$.[2] It is given that $\displaystyle 5\sum_{r=k+1}^{\infty}u_r=9\sum_{r=1}^{k}u_r$. Find the value of $k$.[3]

Accepted Answer

(a) convergent; sum $\frac16$ (b) $u_n=\frac{n}{(n+2)(n+3)(n+4)}$ (c) $k=4$

2026 NYJC P1 Q5

2026 NYJC P1 Q5

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