2026 RI P1 Q2

Junior College 2

6 marks

The sequence $u_1,u_2,u_3,\ldots$ is defined by $u_1=4$ and $u_{n+1}=-\frac13u_n+2$, for $n\geq1$.
1. Write down the value of $u_6$.[1]
2. Given that as $n\to\infty$, $u_n\to l$, find the exact value of $l$ using an algebraic method.[2]
Using the result $\displaystyle\sum_{r=1}^{n}r^2=\frac n6(n+1)(2n+1)$, find $\displaystyle\sum_{r=0}^{n}(n+r)(r+1)$, giving your answer in terms of $n$, in fully factorised form.[3]

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Answer:(a)(i) $\frac{362}{243}$ (ii) $\frac32$ (b) $\frac56n(n+1)(n+2)$

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