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Find the sum : \(\frac{3}{5} + \frac{4}{5^2} + \frac{3}{5^3} + \frac{4}{5^4}\)+ .... To 2n terms

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We can split the above expression into 2 parts. We will split 2n terms into 2 parts also which will leave it as n terms and another n terms.

\(\big(\frac{3}{5} + \frac{3}{5^3} + \) ...... to n terms\(\big)\) + \(\bigg(\frac{4}{5} + \frac{4}{5^2} + .....\) to n terms\(\bigg)\)

Sum of a G.P. series is represented by the formula, Sn = \(a\frac{1-r^n}{1-r}\) , when |r|<1. 

‘Sn’ represents the sum of the G.P. series up to nth terms, 

‘a’ represents the first term, 

‘r’ represents the common ratio and 

‘n’ represents the number of terms. 

Here,

a = \(\frac{3}{5} , \frac{4}{5}\)

r = (ratio between the n term and n-1 term) 

\(\frac{3}{5^2} \div \frac{3}{5}\)\(\frac{4}{5^2} \div \frac{4}{5} = \frac{1}{5^2}, \frac{1}{5}\) n terms

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