Question 0417

Arithmetic and Geometric Progressions (APs, GPs)

2017 Paper 2 Question 2 Variant

Question

An arithmetic progression has first term

{8.}

The sum of the first

{13}

terms of the progression is

{632.}

(i)

Find the common difference.

[2]

A geometric progression has first term

{8}

and common ratio

{r.}

The sum of the first

{13}

terms of the progression is

{632.}

(ii)

Show that

{r^{13} - 79 r + 78=0.}

Show that the common ratio cannot be

{1}

even though

{r=1}

is a root of this equation. Find the possible values of the common ratio.

[4]

(iii)

It is given that the common ratio of the geometric progression is positive, and that the

{n\textrm{th}}

term of this geometric progression is more than

{50}

times the

{n\textrm{th}}

term of the arithmetic progression.

Write down an inequality, and hence find the smallest possible value of

{n.}

[3]