9758/2020/P2/Q02

(a) A sequence is such that $u_{1}=p$, where $p$ is a constant, and $u_{n+1}=2 u_{n}-5$, for $n>0$.

(i) Describe how the sequence behaves when
(A) $p=7$

[1]

(B) $p=5$

[1]

(ii) Find the value of $p$ for which $u_{5}=101$.

[2]

(b) Another sequence is defined by $v_{1}=a, v_{2}=b$, where $a$ and $b$ are constants, and

$v_{n+2}=v_{n}+2 v_{n+1}-7, \quad \text { for } n>0 \text {. }$

For this sequence, $v_{4}=2 v_{3}$.

(i) Find the value of $b$.

[3]

(ii) Find an expression in terms of $a$ for $\nu_{5}$.

[1]

(c) The sum of the first $n$ terms of a series is $n^{3}-11 n^{2}+4 n$, where $n$ is a positive integer.

(i) Find an expression for the $n$th term of this series, giving your answer in its simplest form.

[2]

(ii) 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]