Question

In an infinite geometric progression with positive terms and with a common ratio |r|<1, the sum of the first three terms is 26/3 and the sum of the entire progression is 9. Determine the progression.

Answer 1

given:
a + ar + ar^2 = 26/3
a(1 + r + r^2) = 26/3
a = 26/(3((1 + r + r^2) )

a/(1-r) = 9
a = 9(1-r)

26/(3((1 + r + r^2) ) = 9(1-r)
27(1-r)(1+r+r^2) = 26
27(1 + r + r^2 - r - r^2 - r^3) = 26
27(1 - r^3) = 26
27 - 27r^3 = 26
27r^3 = 1
3r = 1
r = 1/3 , well that went better than expected.

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