If ^ n C ₀+ 1 2 ^ n C ₁+ 1 3 ^ n C ₂+ . . 1 n ^ n C _ n -1 + 1 n +1 ^ n C _ n = 1023 10 then n =

Step-by-step solution

Given the sum S = n 0 + 1 2 n 1 + 1 3 n 2 + + 1 n+1 n n , the general term is 1 r+1 n r . Using the identity 1 r+1 n r = 1 n+1 n+1 r+1 , the sum becomes S = 1 n+1 _ r=0 ^ n n+1 r+1 . Substituting k = r+1 yields S = 1 n+1 _ k=1 ^ n+1 n+1 k . From the binomial theorem, _ k=0 ^ n+1 n+1 k = 2^ n+1 , so _ k=1 ^ n+1 n+1 k = 2^ n+1 - 1 . Thus, S = 2^ n+1 - 1 n+1 . Setting this equal to 1023 10 gives 2^ n+1 - 1 n+1 = 1023 10 . Testing n=9 yields 2¹⁰ - 1 10 = 1023 10 , matching the given value. The correct value is n=9 , co