Find S = 4/7 - 7/7² + 10/7³ - 13/7⁴.........infinity
2 answers
29/64
S = 4/7 - 7/7^2 + 10/7^3 - 13/7^4
multiply both sides by -1/7, and line up the denominators
-1/7S = -4/7^2 + 7/7^3 - 10/7^4 + 13/7^5 - ....
subtract them:
(8/7)S=4/7 - 3/7^2 + 3/7^3 - 3/7^4 .....
= 4/7 - (3/7^2 - 3/7^3 + 3/7^4 - ......)
the part in the bracket is a GP with a = 3/49 and r = -1/7
sum of the terms in the bracket = (3/49) / (1 + 1/7)
= (3/49)(7/8) = 3/56
so (8/7)S = 4/7 - 3/56 = 29/56
S = (29/56)(7/8) = 29/64
multiply both sides by -1/7, and line up the denominators
-1/7S = -4/7^2 + 7/7^3 - 10/7^4 + 13/7^5 - ....
subtract them:
(8/7)S=4/7 - 3/7^2 + 3/7^3 - 3/7^4 .....
= 4/7 - (3/7^2 - 3/7^3 + 3/7^4 - ......)
the part in the bracket is a GP with a = 3/49 and r = -1/7
sum of the terms in the bracket = (3/49) / (1 + 1/7)
= (3/49)(7/8) = 3/56
so (8/7)S = 4/7 - 3/56 = 29/56
S = (29/56)(7/8) = 29/64
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