If $ a_{k} $ is the coefficient of $ x^{k} $ in the expansion of $ \left(1+x+x^{2}\right)^{n} $ for $ k=0,1,2, \ldots ., 2 n $ then value of $ a_{1}+2 a_{2}+3 a_{3}+\ldots \ldots \ldots 2 n a_{2 n} $ is:

Question

If $ a_{k} $ is the coefficient of $ x^{k} $ in the expansion of $ \left(1+x+x^{2}\right)^{n} $ for $ k=0,1,2, \ldots ., 2 n $ then value of $ a_{1}+2 a_{2}+3 a_{3}+\ldots \ldots \ldots 2 n a_{2 n} $ is:, $ -a_{0} $, $ 3^{n} $, $ n \cdot 3^{n+1} $, $ n \cdot 3^{n} $

MARKS App · Accepted Answer

We expand the equation by using binomial expansion, 1 + x + x 2 n = a 0 + a 1 x + a 2 x 3 + .... + a 2 n x 2 n On differentiating both sides, we get n ( 1 + x + x 2 ) n - 1 1 + 2 x = a 1 + 2 a 2 x + 3 a 3 x 2 +... + 2 n a 2 n x 2 n - 1 Now, on putting x = 1 , we get ⇒ n ( 3 ) n - 1 ⋅ 3 = a 1 + 2 a 2 + 3 a 3 + ... + 2 n a 2 n ⟹ a 1 + 2 a 2 + 3 a 3 + .... + 2 n a 2 n = n ⋅ 3 n

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