Question

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $\left( x + \sqrt{x^{3} -1}\right)^{6} + \left( x - \sqrt{x^{3} - 1}\right)^{6}, \left(x > 1\right)$ is equal to :

Answer 1

$\left( x + \sqrt{x^{3} -1}\right)^{6} + \left( x - \sqrt{x^{3} - 1}\right)^{6} $
$ = 2[\,{}^{6}C_{0} x^{6} + \,{}^{6}C_{2}x^{4} \left(x^{3} - 1 \right) + \,{}^{6}C_{4} x^{2} \left(x^{3} -1\right)^{2} $
$+ \,{}^{6}C_{6}\left(x^{3}-1\right)^{3}]$
$ = 2[ x^6 + 15x^7 - 15 x^4 + 15 x^8 - 30 x^5 + 15x^2$
$ + x^9 - 3x^6 + 3x^3 - 1]$
Hence, the sum of coefficients of even powers of
$x = 2[1 -15 + 15 + 15 - 3 - 1] = 24 $