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JEE Main202125 Jul 2021Evening ShiftMathematicsBinomial TheoremActual

Let n ∈ N and x denote the greatest integer less than or equal to x . If the sum of n + 1 terms of C 0 n , 3 · C 1 n , 5 · C 2 n , 7 · C 3 n , … is equal to 2 100 · 101 , then 2 n - 1 2 is equal to

Correct answer

0

Step-by-step solution

Let S = C 0 n + 3 C 1 n + 5 . C 2 n + … + 2 n + 1 · C n n Now, T r = 2 r + 1 n C r Therefore, S = Σ T r S = Σ 2 r + 1 n C r = Σ 2 r n C r + Σ n C r ⇒ S = 2 n · 2 n - 1 + 2 n = 2 n n + 1 ⇒ 2 n n + 1 = 2 100 . 101 ⇒ n = 100 Now, 2 n - 1 2 = 2 99 2 = 98 .

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