Question

The sum of an infinite geometric series is three times the first term. Find the common ratio of this series.

Answer 1

`(2)/(3)`

Explanation:

The sum of an infinite geometric series is expressed according to the formula;

`S_\infty = (a)/(1-r)` where;

a is the first term of the series

r is the common ratio

If the sum of an infinite geometric series is three times the first term, this is expressed as
`S_\infty = 3a`

Substitute
`S_\infty = 3a` into the formula above to get the common ratio r;

`3a = (a)/(1-r) \\\\`

`cross \ multiply\\\\3a(1-r) = a\\\\3(1-r) = 1\\`

open the parenthesis

`3 - 3r = 1\\\\`

subtract 3 from both sides

`3 - 3r -3= 1-3\\\\-3r = -2\\\\r = (2)/(3)`

Hence the common ratio of this series is
`(2)/(3)`