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NTA Abhyas JEE Main2020MathematicsBinomial TheoremPractice

The value of ∑ r = 0 n r + 2 r + 1 C r n is equal to

Options

  1. A2 n n + 2 - 1 n + 1
  2. B2 n n + 1 - 1 n + 1
  3. C2 n n + 4 - 1 n + 1
  4. D2 n n + 3 - 1 n + 1

Correct answer

D. 2 n n + 3 - 1 n + 1

Step-by-step solution

∑ r = 0 n r + 2 r + 1 C r n = ∑ r = 0 n r + 1 + 1 r + 1 C r n = ∑ r = 0 n C r n + ∑ r = 0 n 1 r + 1 C r n = 2 n + ∑ r = 0 n 1 n + 1   n + 1 C r + 1 = 2 n + 1 n + 1 ∑ r = 0 n   n + 1 C r + 1 = 2 n + 1 n + 1 2 n + 1 - 1 = 2 n n + 1 + 2 ⋅ 2 n - 1 n + 1 = 2 n n + 3 - 1 n + 1

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