Vivekanand Vellanki
2

You have to use the binomial theorem to solve this.


The question is: what is the sum of the integral powers of x in

\left(1+2x\sqrt{x}\right)^{40}


Assume: y=2x\sqrt{x}


The equation is \left(1+y\right)^{40}. We are interested in the sum of the coefficients of even powers of y.


Let S be the sum.


S=\sum_{k=0}^{20}\left(^{40}C_{2k}\cdot2^{2k}\right)


The ^{40}C_{2k}term in the summation should be obvious. The 2^{2k} term comes because y=2x\sqrt{x}. When there are 2k y's selected, the 2 in the formula for y contributes to 2^{2k}.