Question 0416

Arithmetic and Geometric Progressions (APs, GPs)

2016 Paper 1 Question 4 Variant

Question

An arithmetic series has first term

{a}

and common difference

{d,}

where

{a}

and

{d}

are non-zero. A geometric series has first term

{b}

and common ratio

{r,}

where

{b}

and

{r}

are non-zero.

It is given that the

{5\textrm{th},}

{10\textrm{th}}

and

{13\textrm{th}}

term of the arithmetic series are equal to the

{2\textrm{nd},}

{5\textrm{th}}

and

{8\textrm{th}}

term of the geometric series respectively.

(i)

Show that

{r}

satisfies the equation

{5 r^6 - 8 r^3 + 3.}

Given that

{|r|<1,}

solve this equation, giving your answer correct to 2 decimal places.

[4]

(ii)

By using this value of

{r,}

find, in terms of

{b}

and

{n,}

the sum of the terms of the geometric series after, but not including, the

{n\textrm{th}}

term, simplifying your answer.

[3]