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Question
Find the sum of the following geometric series:
\[\frac{a}{1 + i} + \frac{a}{(1 + i )^2} + \frac{a}{(1 + i )^3} + . . . + \frac{a}{(1 + i )^n} .\]
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Solution
`a/(1 + i) + a/(1 + i)^2 + a/(1 + i)^3 + ...... + a/(1 + i)^n`
∴ First term, A = `a/(1 + i)`, No. of terms = n,
Common ratio, R = `(a/(1 + i)^2)/(a/(1 + i))`
R = `(cancel(a)/cancel((1 + i))^2)/(cancel(a)/cancel(1 + i))`
∴ R = `1/(1 + i)`
`"S"_"n" = "A" [(1 - "R"^n)/(1 - "R")]`
`= a/(1 + i) [(1 - (1/(1 + i))^n)/(1 - 1/(1 + i))]`
`= a/cancel(1 + i) [(1 - 1/(1 + i)^n)/((cancel(1) + i - cancel(1))/cancel(1 + i))]`
`= a/i xx i/i [1 - (1 + i)^-n]`
`= (ai)/i^2 [1 - (1 + i)^-n]`
`= (ai)/-1 [1 - (1 + i)^-n]`
= - ai [1 - (1 + i)-n]
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