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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

Correct answer

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Step-by-step solution

Given, S = 109 + 108 5 + 107 5 2 + … + 2 5 107 + 1 5 108   . . . . 1 S 5 = 109 5 + 108 5 2 + … + … + 2 5 108 + 1 5 109   . . . . 2 Now subtracting above equations we get, - 4 S 5 = - 109 + 1 5 + 1 5 2 + … + 1 5 108 + 1 5 109 ⇒ - 4 S 5 = - 109 + 1 5 1 - 1 5 109 1 - 1 5 ⇒ 4 S 5 = - 109 + 1 4 1 - 1 5 109 ⇒ 16 S 5 = - 436 + 1 - 1 5 109 ⇒ 16 S = - 436 × 5 + 5 - 5 - 108 ⇒ 16 S - ( 25 ) - 54 = 2175

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