Question

The sum of first three terms of a G.P. is 13/12 and their product is -1. Find the G.P.?

Answer 1

The geometric progression is `4/3, -1, 3/4` or `3/4, -1, 4/3`.

Explanation:

Recall that the next term in a geometric progression is given by muliplying the previous term by a common ratio `r`.

Therefore, if the first term is `a_1`

`a_2 = a_1r`
`a_3 = a_1(r)(r) = a_1r^2`

Therefore:

`a_1 + a_1r + a_1r^2 = 13/12`
`a_1(a_1r)(a_1r^2) = -1`

Simplifying the equations a little, we get:

`a_1(1 + r + r^2) = 13/12`

`a_1^3 = -1/r^3`
`a_1 = -1/r`

We substitute this into the first equation to get

`-1/r(1 + r + r^2) = 13/12`

`(-1 - r - r^2)/r = 13/12`

`-12 - 12r - 12r^2 = 13r`

`0 = 12r^2 + 25r + 12`

This can be readily solved by factoring. Two numbers that multiply to `144` and add to `25` are `16` and `9`.

`0 = 12r^2 + 16r + 9r + 12`

`0 = 4r(3r + 4) + 3(3r + 4)`

`0 = (4r + 3)(3r + 4)`

`r = -3/4 or -4/3`

Since

`a_1 = -1/r`, we get that `a_1 = -1/(-3/4) = 4/3` or `a_1 = -1/(-4/3) = 3/4`

It follows that for the first progression (with common ratio `-3/4`)) the second term is `-1` and the third term is `3/4`.

For the second progression (with common ratio `-4/3`), the second term is `-1` and the third term is `4/3`.

Hopefully this helps!