A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where

Question

A sequence ${{u}_{1}}$, ${{u}_{2}}$, ${{u}_{3}}$ ... is such that ${{u}_{n+1}}=3{{u}_{n}}+Pn$, where $P$ is a constant and $n\in {{\mathbb{Z}}^{+}}$. The terms of the sequence are defined by their previous terms, for example, ${{u}_{2}}=3{{u}_{1}}+P$. Given that ${{u}_{1}}=1$ and ${{u}_{2}}=7$, find $P$ and ${{u}_{3}}$. [2] lt is known that the ${{n}^{\mathrm{th}}}$ term of this sequence is given by ${{u}_{n}}=a\left( {{3}^{n}} \right)+bn+c$, where $a$, $b$ and $c$ are constants. Find $a$, $b$ and $c$. [4] Find $\sum\limits_{r=1}^{n}{{{u}_{r}}}$ in terms of $n$. [4]

Accepted Answer

(i) $P=4$, $u_3=29$ (ii) $a=\dfrac43$, $b=-2$, $c=-1$ (iii) $\displaystyle\sum_{r=1}^{n}u_r=2(3^n-1)-n(n+2)$

2020 CJC Promo Q9

2020 CJC Promo Q9

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