An infinite geometric progression has positive first term and common ratio

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

Problem

An infinite geometric progression has positive first term

a

and common ratio

r

. The sum to infinity of the series is

12

, while the sum of its first three terms is

10

. (i) Write expressions for the sum to infinity and for the sum of the first three terms of the series in terms of

a

and

r

. (ii) Using the given numerical values, form two equations in

a

and

r

. (iii) Solve the equations to find the values of

a

and

r

, and comment on whether the obtained

r

satisfies the convergence condition.