Question 0407

Arithmetic and Geometric Progressions (APs, GPs)

2007 Paper 1 Question 10 Variant

Question

A geometric series has common ratio

{r,}

and an arithmetic series has first term

{a}

and common difference

{d,}

where

{a}

and

{d}

are non-zero. The first three terms of the geometric series are equal to the first, eighth and ninth terms respectively of the arithmetic series.

(i)

Show that

{7 r^2 - 8 r + 1=0.}

[4]

(ii)

Deduce that the geometric series is convergent and find, in terms of

{a,}

the sum to infinity.

[5]

(iii)

The sum of the first

{n}

terms of the arithmetic series is denoted by

{S.}

Given that

{a>0,}

find the set of possible values of

{n}

for which

{S}

exceeds

{2a.}

[5]