The sum of three numbers in G.P. is 13/12 and their product is -1. Which are these numbers ?

If the 3 numbers are the consecutive terms of a geometric
progression, we'll note them as:

a/r , a ,
a*r

From enunciation, we know that the product of the
numbers is:

(a/r) * (a) * (a*r) =
-1

We'll eliminate like
terms:

a*a*a = -1

a^3 = -1
=> a = -1

So, the numbers
are:

-1/r , -1 , -1*r

Also
from enunciation, we know that:

(-1/r) +( -1) + (-1*r) =
13r/12

To add the 3 numbers, we'll calculate LCD,which is
r:

-1 -r - r^2 = 13r/12

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

We'll move all terms to one
side:

12r^2 + 12r + 12 + 13r =
0

12r^2 + 25r + 12 = 0

We'll
apply the quadratic formula:

r1 =
[-25+sqrt(625-576)]/24

r1 =
(-25+7)/24

r1 = -18/24

r1 =
-3/4

r2 = (-25-7)/24

r2 =
-32/24

r2 = -4/3

So, when the
common ratio is r = -3/4, the geometric progression is:

4/3
, -1 , 3/4

When the common ratio is r = -4/3, the G.P.
is:

3/4 , -1 , 4/3