frac{2}{5} + frac{3}{5^{2}} + frac{2}{5^{3}} | vedclass

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(C) The given series is $\frac{2}{5} + \frac{3}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + \dots \infty$.We can split this into two separate infinit

Vedclass · Accepted Answer

$\frac{13}{24}$

Vedclass · Answer

(C) The given series is $\frac{2}{5} + \frac{3}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + \dots \infty$.We can split this into two separate infinite geometric series:$S = \left( \frac{2}{5} + \frac{2}{5^{3}} + \frac{2}{5^{5}} + \dots ight) + \left( \frac{3}{5^{2}} + \frac{3}{5^{4}} + \frac{3}{5^{6}} + \dots ight)$.For the first series,the first term $a_1 = \frac{2}{5}$ and the common ratio $r = \frac{1}{5^{2}} = \frac{1}{25}$.The sum is $S_1 = \frac{a_1}{1 - r} = \frac{2/5}{1 - 1/25} = \frac{2/5}{24/25} = \frac{2}{5} 	imes \frac{25}{24} = \frac{5}{12}$.For the second series,the first term $a_2 = \frac{3}{5^{2}} = \frac{3}{25}$ and the common ratio $r = \frac{1}{25}$.The sum is $S_2 = \frac{a_2}{1 - r} = \frac{3/25}{1 - 1/25} = \frac{3/25}{24/25} = \frac{3}{24} = \frac{1}{8}$.Total sum $S = S_1 + S_2 = \frac{5}{12} + \frac{1}{8} = \frac{10 + 3}{24} = \frac{13}{24}$.

$\frac{2}{5} + \frac{3}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + \dots \infty$

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