Two infinite geometric series and satisfy the following information

MATHEMATICS · Multi-part · CBSE · Mathematics · Sequences and Series · Sequences and Series

Problem

Two infinite geometric series

S_1

and

S_2

satisfy the following information. • The first term of

S_1

10

and its common ratio is

\dfrac13

. • The ratio of the sums to infinity is

S_1:S_2=3:2

. • The first term of

S_2

4

. Answer the following questions. (i) Find the sum to infinity of

S_1

. (ii) Using the given ratio of sums, determine the exact value of the sum to infinity of

S_2

. (iii) Hence find the common ratio of

S_2

. (iv) Verify whether

S_2

converges with the common ratio obtained in part (iii).