The first term of a G.P. is 1. The sum of the third and fifth term is 90. Calculate the common ratio of the G.P.

We'll note the first term of the G.P. as a1 = 1 and the
common ratio as r.

From the enunciation, we know
that:

a3 + a5 = 90

From the
standard formula of the n-th term of the G.P., we have:

an
= a1*r^(n-1)

Now, we can calculate a3 and
a5:

a3 = a1*r^2 = r^2

a5 =
a1*r^4 = r^4

a3 + a5 = 90 => r^2 +  r^4 =
90

We'll substitute r^2 =
t

t^2 + t - 90 = 0

We'll apply
the quadratic formula:

t1 =
[-1+sqrt(1+360)]/2

t1 =
(-1+19)/2

t1 = 9

t2 =
(-1-19)/2

t2 = -10

But r^2 =
t

So, r^2 = t1

r^2 =
9

r1 = +3

r2 =
-3

r^2 = t2

r^2 = -10
impossible

So, the common ratio could be r = -3 or r =
3.