Let S = 109 + frac{108}{5} + frac{107}{5^2} + | vedclass

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(B) Given $S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{1}{5^{108}}$.Multiply by $\frac{1}{5}$: $\frac{S}{5} = \frac{109}{5} + \frac{108

Vedclass · Accepted Answer

$2175$

Vedclass · Answer

(B) Given $S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{1}{5^{108}}$.Multiply by $\frac{1}{5}$: $\frac{S}{5} = \frac{109}{5} + \frac{108}{5^2} + \ldots + \frac{2}{5^{108}} + \frac{1}{5^{109}}$.Subtracting the two equations:$S - \frac{S}{5} = 109 + (\frac{108-109}{5}) + (\frac{107-108}{5^2}) + \ldots + (\frac{1-2}{5^{108}}) - \frac{1}{5^{109}}$.$\frac{4S}{5} = 109 - (\frac{1}{5} + \frac{1}{5^2} + \ldots + \frac{1}{5^{108}}) - \frac{1}{5^{109}}$.$\frac{4S}{5} = 109 - [\frac{1}{5} \frac{(1 - (1/5)^{108})}{(1 - 1/5)}] - \frac{1}{5^{109}}$.$\frac{4S}{5} = 109 - [\frac{1}{5} \cdot \frac{5}{4} (1 - \frac{1}{5^{108}})] - \frac{1}{5^{109}}$.$\frac{4S}{5} = 109 - \frac{1}{4} (1 - \frac{1}{5^{108}}) - \frac{1}{5^{109}}$.Multiply by $\frac{5}{4}$:$S = \frac{5}{4} [109 - \frac{1}{4} + \frac{1}{4 \cdot 5^{108}} - \frac{1}{5^{109}}]$.$16S = 20 \cdot 109 - 5 + \frac{5}{5^{108}} - \frac{4}{5^{108}} = 2180 - 5 + \frac{1}{5^{108}}$.Since $(25)^{-54} = (5^2)^{-54} = 5^{-108} = \frac{1}{5^{108}}$.$16S - (25)^{-54} = 2175 + \frac{1}{5^{108}} - \frac{1}{5^{108}} = 2175$.

Let $S = 109 + \frac{108}{5} + \frac{107}{5^2} + \ldots + \frac{2}{5^{107}} + \frac{1}{5^{108}}$. Then the value of $(16S - (25)^{-54})$ is equal to $............$.

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