SOLUTION: The first three terms of geometric progression are x+1, x-3 and x-1. find i) the value of x ii) the first term iii) the sum to infinity I enjoy Edwin's explanation

Question 1105079: The first three terms of geometric progression are x+1, x-3 and x-1. find
i) the value of x
ii) the first term
iii) the sum to infinity
I enjoy Edwin's explanation
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
the general form of a geometric progression(GP) is
:
x(n) = ar^(n-1) with a = first term, r is the common ratio
:
we have a = x + 1, then
:
x(1) = x + 1
x(2) = x - 3 = (x + 1)r
x(3) = x - 1 = (x + 1)r^2
:
solve x(2) for r
:
r = (x - 3) / (x + 1)
:
solve x(3) for r^2
:
r^2 = (x - 1) / (x + 1)
:
then, we have
:
(x - 1) / (x + 1) = (x - 3)^2 / (x + 1)^2
:
cross multiply the fractions
:
(x - 1) * (x + 1)^2 = (x + 1) * (x - 3)^2
:
(x - 1) * (x + 1) = (x - 3)^2
:
x^2 -1 = x^2 -6x +9
:
6x = 10
x = 10/6 = 5/3
:
x(1) = a = 5/3 + 1 = 8/3
r = (5/3 - 3) / (5/3 + 1) = (-4/3) / (8/3) = -1/2
:
x(2) = 5/3 - 3 = -4/3
:
X(3) = 5/3 - 1 = 2/3
:
GP is 8/3, -4/3, 2/3
:
now |r| = 1/2 < 1, so we know the infinite sum of this GP converges
:
convergent sum = a / 1 - r = (8/3) / (1 - (-1/2)) = 8/3 * 2/3 = 16/9
:

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