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Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration

Let u_n= ₀^ / 2 ^n x n x d x it turns out that u₁, u₂, u₃, . form a G.P. then

Options

  1. Athe common ratio of G.P. is 1 2
  2. Bthe common ratio of G.P. is 1 4
  3. Cthen u_n= 2^ 2 n
  4. Dthen u_n= 2^ n+1

Correct answer

D. then u_n= 2^ n+1

Step-by-step solution

First use integrating by parts to show that aligned & u_n= ₀^ / 2 ^ n-1 x x n x d x & = ₀^ / 2 ^ n-1 x (n-1) x- ₀^ / 2 ^n x n x d x( Use n x x= (n-1) x- n x x) & =u_ n-1 -u_n & u_n u_ n-1 = 1 2 aligned We have, u₁= 4 , u_n= 1 2^ n-1 4 = 2^ n+1

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