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NDA2023MathematicsBinomial TheoremActual

Let (1+x)^n=1+T₁ x+T₂ x^2+T₃ x^3+ +T_n x^n What is T₁+2 T₂+3 T₃+ +n T_n equal to?

Options

  1. A0
  2. B1
  3. C2^ n
  4. Dn 2^ n-1

Correct answer

D. n 2^ n-1

Step-by-step solution

aligned & Given, (1+ x )^ n =1+ T ₁ x + T ₂ x ^2+ .+ T _ n x ^ n & differentiating both side with respect to x , we get & n (1+ x )^ n -1 = T ₁+2 ~T ₂ x ^1+2 ~T ₃ x ^2+ .+ nT _ n x ^ n -1 ( i ) & n (1+1)^ n -1 = T ₁+2 ~T ₂+3 ~T ₃+ .+ nT _ n & . T ₁+2 ~T ₂+3 ~T ₃+ .+ nT _ n = n ^ n -1 Putting x =1 aligned

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