# The sum of the first two terms of a geometric sequence is #8/9# and of the first three terms is #26/27#. What is the first term, common ratio and sum to infinity ?

##### 1 Answer

Aug 14, 2016

The first term can be

The first term can be

These are the only two possibilities.

#### Explanation:

If the first term is

#a(1+r) = 8/9#

#a(1+r+r^2) = 26/27#

So:

#13/12 = (26*9)/(27*8) = (color(red)(cancel(color(black)(a)))(1+r+r^2))/(color(red)(cancel(color(black)(a)))(1+r)) = (1+r+r^2)/(1+r)#

Cross multiply to get:

#13(1+r) = 12(1+r+r^2)#

Rearrange to get:

#0 = 12r^2-r-1 = (3r-1)(4r+1)#

So:

#r = 1/3 " "# or#" " r = -1/4#

If

The sum of the whole series is then

If

The sum of the whole series is then