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AP EAMCET2012MathematicsBinomial Theorem

_ k=1 ^ _ r=0 ^k 1 3^k ( ^k C_r ) is equal to

Options

  1. A1 3
  2. B2 3
  3. C1
  4. D2

Correct answer

D. 2

Step-by-step solution

aligned & _ k=1 ^ _ r=0 ^k 1 3^k ( ^k C_r ) = & _ k=1 ^ 1 3^k ( ^k C₀+ ^k C₁+ ^k C₂+ + ^k C_k ) = & _ k=1 ^ 2^k 3^k = _ k=1 ^ ( 2 3 )^k = & 2 3 + ( 2 3 )^1+ = & 2 / 3 1- 2 3 ( common ratio = 2 3 ) = & 2 aligned

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