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If (18 x^2+12 x+4)^n=a0+a(1x)+a2x2++a(2n...

If `(18 x^2+12 x+4)^n=a_0+a_(1x)+a2x2++a_(2n)x^(2n),` prove that `a_r=2^n3^r(^(2n)C_r+^n C_1^(2n-2)C_r+^n C_2^(2n-4)C_r+)` .

Text Solution

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`(18x^(2)+12x+4)^(n) = 2^(n)[2+9x^(2)+6x]^(n)`
Now, `a_(r )` is coefficient of `x^(r )` in `2^(n) [(3x+1)^(2)+1]^(n)`. Hence
`a_(r) =` Coefficient of `x^(r )2^(n)[.^(n)C_(0)(3x+1)^(2n)+.^(n)C_(1)(3x+1)^(2n-2) + .^(n)C_(2)(3x+1)^(2n-4)+"…."+.^(n)C_(r )(3x+1)^(2n-2r)+"....."]`
or `a_(r)=2^(n)[.^(n)C_(0)3^(r).^(2n)C_(r)+.^(n)C_(1)3^(r).^(2n-2)C_(r)+.^(n)C_(2)3^(r).^(2n-4)C_(r)+"...."]`
`= 2^(n)3^(r)[.^(n)C_(0).^(2n)C_(r)+.^(n)C_(1).^(2n-2)C_(r)+.^(n)C_(2).^(2n-4)C_(r)+"...."]`
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