Question

The second term of a geometric progression is 13 more than the first term given that the common ratio is half the first term. Find the 3rd term of the G.P.

Answer 1

ar - a = 13 and r = a/2

a(a/2) - a = 13
a^2 - 2a - 26 = 0

a = 1 ± 3√3

if a = 1 + 3√3, then r = (1+3√3)/2
and term(3) = ar^2 = (1+3√3) (1+ 3√3)^2 / 4

if a = 1 - 3√3, then r = (1-3√3)/2
and term(3) = ar^2 = (1-3√3) (1- 3√3)^2 / 4 , I will let you evaluate those

but checking
terms are : 1+3√3, (1+3√3)^2 / 2

difference between 2nd and first = (1+3√3)^2 / 2 - (1+3√3)
= (1 + 6√3 + 27)/2 - 1 - 3√3
= (28 + 6√3)/2 -1 - 3√3
= 14 + 3√3 - 1 - 3√3
= 13 , as required