A sequence is such that ${{u}_{1}}=p$ where $p$ is a constant and \({{u}_{n+1}}=2{{u}_

Question

A sequence is such that ${{u}_{1}}=p$ where $p$ is a constant and ${{u}_{n+1}}=2{{u}_{n}}-5$, for $n>0$. Describe how the sequence behaves when (A) $p=7$,[1] (B) $p=5$.[1] Find the value of $p$ for which ${{u}_{5}}=101$.[2] Another sequence is defined by ${{v}_{1}}=a$, ${{v}_{2}}=b$, where $a$ and $b$ are constants, and ${{v}_{n+2}}={{v}_{n}}+2{{v}_{n+1}}-7$, for $n>0$. For this sequence, ${{v}_{4}}=2{{v}_{3}}$. Find the value of $b$.[3] Find an expression in terms of $a$ for ${{v}_{5}}$.[1] The sum of the first $n$ terms of a series is ${{n}^{3}}-11{{n}^{2}}+4n$, where $n$ is a positive integer. Find an expression for the $n$th term of this series, giving your answer in its simplest form.[2] The sum of the first $m$ terms of this series, where $m>3$, is equal to the sum of the first three terms of this series. Find the value of $m$.[2]

Accepted Answer

(a)(ii) ${11}$ (b)(i) ${7}$ (ii) ${5a+28}$ (c)(i) ${3n^2-25n+16}$ (ii) ${10}$

N2020 P2 Q2

N2020 P2 Q2

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