Question 0520

Sigma Notation

2020 Paper 2 Question 2 Variant

Question

(a)

A sequence is such that

{u_1=p,}

where

{p}

is a constant, and

u_{n+1} = 4 u_n - 3, \quad \textrm{for } n > 0.

(ai)

Describe how the sequence behaves when

{\textrm{(A) } p=5,}

{\textrm{(B) } p=1.}

[2]

(aii)

Find the value of

{p}

for which

{u_5 = 769.}

[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 + 4 v_{n+1} - 2, \quad \textrm{for } n > 0.

For this sequence,

{v_4 = 4v_3.}

(bi)

Find the value of

{b.}

[3]

(bii)

Find an expression in terms of

{a}

for

{v_5.}

[1]

(c)

The sum of the first

{n}

terms of a series is

{n^3 - 12 n^2 + 5 n,}

where

{n}

is a positive integer.

(ci)

Find an expression for the

{n\textrm{th}}

term of this series, giving your answer in its simplest form.

[2]

(cii)

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]