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Question: The greatest coefficient in the expansion of \((1 + 3\sqrt{2}x)^{9} + (1 - 3\sqrt{2}x)^{9}\) is....

The greatest coefficient in the expansion of (1+32x)9+(1−32x)9(1 + 3\sqrt{2}x)^{9} + (1 - 3\sqrt{2}x)^{9} is.

A

73007^{300}

B

(2x2−13x2)10\left( 2x^{2} - \frac{1}{3x^{2}} \right)^{10}

C

66

D

458017\frac{4580}{17}

Answer

73007^{300}

Explanation

Solution

(1+x)2n+2(1 + x)^{2n + 2}

Here, N = 2n +1 ⇒ (2n)!(n!)2\frac{(2n)!}{(n!)^{2}}

(2n+2)!{(n+1)!}2\frac{(2n + 2)!}{\{(n + 1)!\}^{2}} (2n+2)!n!(n+1)!\frac{(2n + 2)!}{n!(n + 1)!}

(2n)!n!(n+1)!\frac{(2n)!}{n!(n + 1)!} 3(1+13)20\sqrt{3}\left( 1 + \frac{1}{\sqrt{3}} \right)^{20} 258409\frac{25840}{9} 248409\frac{24840}{9}⇒ 268409\frac{26840}{9}

(1+x)n(1 + x)^{n}

Tr+1=Tn+1=2n+1Cn+1T _ { r + 1 } = T _ { n + 1 } = { } ^ { 2 n + 1 } C _ { n + 1 } n+1n<x<nn+1\frac{n + 1}{n} < x < \frac{n}{n + 1}.