Geometric Progression?

Question

The first three terms of an infinite geometric progression, for which all the terms are positive, are 8x+1, 2x+7 and x-1 respectively.
Find:
(i) the value of x;
(ii) the sum to infinity of the progression.

Answer 1

A geometric progression is one in which each term is equal to the previous term, multiplied by the same constant factor. Therefore, we can state that
(x-1)/(2x+7) = (2x+7)/(8x+1)

The ratio of the 3rd to the 2nd term must be the same as the ratio of the 2nd to the 1st term. Solving for x, we find that
x = -5/4, or
x = 10

If we try x = -5/4, we get negative numbers for terms in the series, so x must be 10, which gives us
8x+1, 2x+7, x-1...
= 81, 27, 9...

We can see the nth term in the series is just
3^(-n + 5), so the series would go
81, 27, 9, 3, 1, 1/3, 1/9, 1/27, 1/81, 1/243...

The sum of all terms from n=1 to n=infinity is 243/2.

Answer 2

(2x+1)^2 =(8x +1)(x-1) then 4x^2-35x -50=0 thenx 10 or -1.25refused as it is + ive term , then x= 10
then 81 , 27 , 9 . r =1/3 then S∞ = a/(1-r) = 81/1-(1/3) =81*3/2 =243/2 =121.5