a1, a2, a3, ..... is a geometric progression.

We know that:

a1= 1

a3 + a5 = 90

But:

a3= a1*r^2 = r^2

a5 = a1 *r^4 = r^4

==> r^2 + r^4 = 90

==> r^4 + r^2 - 90 =0

==> (r^2 + 10) (r^2 - 9) = 0

==> (r^2 + 10) ( r-3)(r+3) = 0

==> r1= 3

==> r2= -3