2026 RVHS P1 Q8

Junior College 2

8 marks

A sequence $u_1,u_2,u_3,\ldots$ is such that $u_1=5$ and $u_{n+1}=3u_n+2n$.
It is known that the $n$th term of this sequence is given by
\[u_n=a(3^n)+bn+c,\]
where $a$, $b$ and $c$ are constants.
Find the constants $a$, $b$ and $c$.[4]
1. State the value of $k$ such that $\displaystyle\sum_{r=1}^{2n}\frac1r=\sum_{r=1}^{n}\left(\frac1{2r+k}+\frac1{2r}\right)$.[1]
2. Hence, show that $\displaystyle\sum_{r=n+1}^{2n}\frac1r=\sum_{r=1}^{n}\left(\frac1{2r-1}-\frac1{2r}\right)$.[2]
3. Write down the value of $\displaystyle\lim_{n\to\infty}\sum_{r=n+1}^{2n}\frac1r$.[1]

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Answer:(a) $a=\frac{13}6,\ b=-1,\ c=-\frac12$ (b)(i) $k=-1$ (ii) Shown (iii) $\ln2$

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