If the value of C₀+2 C₁+3 C₂+ +(n+1) C_ n =576 , then n is equal to

Step-by-step solution

Given, C₀+2 C₁+3 C₂+ +(n+1) C_ n =576 We know that, array r (1+x)^ n = n 0 + n 1 x+ n 2 x²+ + n n x^ n x(1+x)^ n = n 0 x+ n 1 x²+ n 2 x³+ + n n x^ n+1 array On differentiating w.r.t. x , we get .= n 0 +2 n 1 x+3 )^ n C₂ x²+ +(n+1) n n x^ n On putting n=1 , we get 2^ n +n 2^ n-1 = n 0 +2 n 1 +3 n 1 + +(n+1) n n (given) 2^ n-1 (n+2)=2⁶ 9=2^ (7-1) (7+2) On comparing, we get n=7