If a_k is the coefficient of x^k in the expansion of (1+x+x^2 )^n for k=0,1,2, , 2 n , then a₁+2 a₂+3 a₃+ +2 n a_ 2 n is equal to

Step-by-step solution

We have, aligned (1+x+x^2 )^n=a₀+a₁ x+a₂ x^2+a₃ x^3 & + & +a_ 2 x^ 2 n aligned On differentiating both sides, we get array r n (1+x+x^2 )^ n-1 (1+2 x)=a₁+2 a₂ x+3 a₃ x + +2 n a_ 2 n x^ 2 n-1 array Now, putting x=1 , we get aligned & n(3)^ n-1 (3)=a₁+2 a₂+3 a₃+ +2 n a_ 2 n & a₁+2 a₂+3 a₃+ +2 n a_ 2 n =n 3^n aligned