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JEE Main202124 Feb 2021Evening ShiftMathematicsBinomial TheoremActual

If n ⩾ 2 is a positive integer, then the sum of the series C 2 n + 1 + 2 C 2 2 + C 2 3 + C 2 4 + … + C 2 n is

Options

  1. An n - 1 2 n + 1 6
  2. Bn n + 1 2 n + 1 6
  3. Cn n + 1 2 n + 2 12
  4. Dn 2 n + 1 3 n + 1 6

Correct answer

B. n n + 1 2 n + 1 6

Step-by-step solution

C 2 n + 1 + 2 C 2 2 + C 2 3 + C 2 4 + … + C 2 n C 2 n + 1 + 2 C 3 3 + C 2 3 + C 2 4 + … + C 2 n use C r + 1 n + C r n = C r n + 1 = C 2 n + 1 + 2 C 3 4 + C 2 4 + C 3 5 + … + C 2 n = C 2 n + 1 + 2 C 3 5 + C 2 5 + … + C 2 n = C 2 n + 1 + 2 C 3 n + C 2 n = C 2 n + 1 + 2 · C 3 n + 1 = n + 1 n 2 + 2 · n + 1 n n - 1 2 . 3 = n n + 1 2 n + 1 6

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