In a geometric progression ,the sum of 3rd and 4th term is 108. if the sum of the 4th and 5th term is 324, determine the: common ratio, first term and the 12th term.

see explanation

Step-by-step explanation:

The terms of a geometric progression are

a, ar, ar², ar³, ...... , a

Thus the sum of the third and fourth is

ar² + ar³ = 108 → (1)

The sum of the fourth and fifth is

ar³ + a = 324 → (2)

Factorise both equations

ar²(1 + r) = 108 → (3)

ar³(1 + r) = 324 → (4)

Divide (4) by (3)

=

Cancelling (1 + r) , a and r, gives

r = 3

Substitute r = 3 into (3) and solve for a

9a(4) = 108

36a = 108 ( divide both sides by 36 )

a = 3

The n th term is

= a with a = 3 and r = 3, hence

= 3 = 531441