Question

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

Answer 1

a1= 1

a3 + a5 = 90

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

r=+3,-3

Answer 2

Hi there!

$a_1$, $a_2$, $a_3$, ..... is a Geoometric progression.

Given,

$a_1$ = 1

$a_3$ + $a_5$ = 90

But_

$a_3$ = $a_1$ r² = r²

$a_5$ = $a_1$ r⁴ = r⁴

r² + r⁴ = 90

r⁴ + r⁴ - 90 = 0

(r2 + 10) (r² - 9) = 0

(r² + 10) ( r - 3)(r + 3) = 0

Thus,

$r_1$ = 3

$r_2$ = -3

Hope it helps! :D

Cheers,
Ishaan Singh