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JEE Main202224 Jun 2022Evening ShiftMathematicsBinomial TheoremActual

The remainder on dividing 1 + 3 + 3 2 + 3 3 + … + 3 2021 by 50 is _____.

Correct answer

0

Step-by-step solution

Let S = 1 + 3 + 3 2 + ⋯ · 3 2021 = 3 2022 - 1 3 - 1 = 3 2022 - 1 2 Simplifying by using binomial expansion we get 3 2 1011 - 1 2 = ( 10 - 1 ) 1011 - 1 2 ⇒   1011 C 0 10 1011 - 1011 C 1 10 1010 + … + C 1010 1011 10 1 - C 1011 1011 × 1 - 1 2 100 q + 1011 × 10 - 2 2 ⇒     100 q + 10110 - 2 2 = 100 q + 10108 2 = 50 q + 5054 Now dividing 50 q + 5054 by 50 We get 4 as remainder. Hence option B is correct.

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