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Question
Find the sum to indicated number of terms of the geometric progressions `sqrt7, sqrt21,3sqrt7`...n terms.
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Solution
The geometric series `sqrt7, sqrt21, 3sqrt7,...`
First term, a = `sqrt7`
Common ratio, r = `sqrt21/sqrt7 = sqrt3`
Sum of n terms = `("a"(1 - "r"^"n"))/(1 - "r")` when r > 1
= `(sqrt7 [(sqrt3)^"n" - 1])/("r" -1)`
= `(sqrt7 [(sqrt3)^("n"/2) - 1])/(sqrt3 -1) xx (sqrt3 + 1)/(sqrt3 + 1)`
= `(sqrt7 (sqrt3 + 1)(3^("n"/2) - 1))/2`
= `(sqrt7(1 + sqrt3))/2[(3)^"n"/2 - 1]`
= `(sqrt7(1 + sqrt3))/2[(3)^(n/2) - 1]`
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