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AP EAMCET2001MathematicsBinomial Theorem

If (1+x)^n=C₀+C₁ x+C₂ x^2+ +C_n x^n , then C₀+2 C₁+3 C₂+ +(n+1) C_n is equal to

Options

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

Correct answer

A. 2^n+n 2^ n-1

Step-by-step solution

Let S=C₀+2 C₁+3 C₂+ +(n+1) C_n (i) aligned & S=C_n+2 C_ n-1 +3 C_ n-2 + +(n+1) C₀ & or S=(n+1) C₀+n C₁+ +C_n aligned On adding Eqs. (i) and (ii), we have aligned 2 S & =C₀[1+n+1]+C₁[2+n]+ +C_n[n+1+1] 2 S & =(n+2) (C₀+C₁+ +C_n ) S & = (n+2) 2 2^n=(n+2) 2^ n-1 =n 2^ n-1 +2^n aligned

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