Question

The eight,fourth and second terms of an arithmetic progression form the first three terms of a geometric series. The arithmetic progression has first term A and common difference d, and the geometric progression has first term G and common ratio r.
a)Given that d is not equal to 0,find the value of r
b)Given that A=2,find the sum to infinity of the geometric progression

Answer 1

8th term of AS = a+7d
4th term of AS = a+3d
2nd term of AS = a+d

then, if they form the first three terms of a GS
(a+3d)/(a+7d) = (a+d)/(a+3d)
(a+3d)^2 = (a+7d)(a+d)
a^2 + 6ad + 9d^2 = a^2 + 8ad + 7d^2
2d^2 - 2ad = 0
d(d-a) = 0
d = 0 or d = a
but we are told that d ≠ 0 , so
d = a

r = (a+3d)/(a+7d)
= 4a/(8a) = 1/2

f) if a = 2
sum(all terms) = a/(1-r)
= 2/(1 - 1/2)
= 4