Consider two geometric progressions

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider two geometric progressions, each consisting of

n

terms and each having the same positive first term

a

. The first progression has common ratio

r_{1}

where

0<r_{1}<1

, while the second has common ratio

r_{2}

where

r_{2}>1

. (i) Write the expressions for the sums

S_{1}

and

S_{2}

of the two progressions in terms of

a, r_{1}, r_{2}

and

n

. (ii) Obtain an expression for

S_{2}-S_{1}

in its simplest form. (iii) Keeping

a, r_{1}

and

r_{2}

fixed, explain with reasons how the value of

S_{2}-S_{1}

changes as

n

increases.