Let S = 109 + 108 5 + 107 5 2 + . . . + 2 5 107 + 1 5 108 . Then the value of ( 16 S - ( 25 ) - 54 ) is equal to _________ . [2023]

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Solution

Q.
Let $S = 109 + \frac{108}{5} + \frac{107}{5^{2}} + . . . + \frac{2}{5^{107}} + \frac{1}{5^{108}}$ . Then the value of $(16 S - {(25)}^{- 54})$ is equal to _________ . [2023]

Ans.
(2175)

$S = 109 + \frac{108}{5} + \frac{107}{5^{2}} + \dots + \frac{2}{5^{107}} + \frac{1}{5^{108}}$ ...(i)

$\frac{S}{5} = \frac{109}{5} + \frac{108}{5^{2}} + \dots + \frac{2}{5^{108}} + \frac{1}{5^{109}}$ ...(ii)

Subtracting (ii) from (i), we get

$\frac{4 S}{5} = 109 - \frac{1}{5} - \frac{1}{5^{2}} \dots - \frac{1}{5^{108}} - \frac{1}{5^{109}}$

$= 109 - (\frac{1}{5} \frac{(1 - \frac{1}{5^{109}})}{(1 - \frac{1}{5})}) = 109 - \frac{1}{4} (1 - \frac{1}{5^{109}})$

$= 109 - \frac{1}{4} + \frac{1}{4} \times \frac{1}{5^{109}} \Rightarrow S = \frac{5}{4} (109 - \frac{1}{4} + \frac{1}{4 \cdot 5^{109}})$

$\Rightarrow 16 S = 20 \times 109 - 5 + \frac{1}{5^{108}}$

$∴$ $16 S - {(25)}^{- 54} = 2180 - 5 = 2175$

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