An infinite geometric progression has first term and common ratio with

MATHEMATICS · Subjective · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

An infinite geometric progression has first term

a

and common ratio

r

with

|r|<1

. (i) Establish that the remainder after

n

terms,

R_{n}=S_{\infty}-S_{n}

, is given by

\dfrac{a r^{n}}{1-r}

. (ii) For

a=7

and

r=\dfrac23

, determine the smallest positive integer

n

such that

R_{n}<0.01

. (iii) Discuss, with reasons, how the magnitude of

r

(while still satisfying

|r|<1

) affects the rate at which

S_{n}

approaches

S_{\infty}