Let be the first term and the common ratio of a geometric progression

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

Problem

Let

a

be the first term and

r\,(\neq1)

the common ratio of a geometric progression. The sum of the first

n

terms is denoted by

S_n

. (i) Show that

S_{2n}=S_n+r^{\,n}\,S_n

. (ii) Hence obtain

\displaystyle\frac{S_{2n}}{S_n}

in terms of

r

and

n

. (iii) Taking

r=2

, determine the least positive integer

n

for which

S_{2n}

exceeds

S_n

by more than

999a

. Show all steps clearly.