Question

Let $2{(1+x^3)}^{100}=\sum _{i=0}^{100}\{a_i{x}^i-\cos \left(\frac{\pi }{2}\right(x+i\left)\right)\}.$ If $\sum _{i=0}^{50}{a}_{2i}=2^k,$ then the value of $k$ is

Answer 1

$2(1+x^3{)}^{100}=[a_0+a_{1}x+a_2{x}^2+\cdots +a_{100}{x}^{100}]-[\cos \frac{\pi }{2}x+\cos (\frac{\pi }{2}x+\frac{\pi }{2})+\cdots +\cos (\frac{\pi }{2}x+\frac{100\pi }{2})]\cdots (1)$

Replacing $x$ by $-x,$ we get
$2(1-x^3{)}^{100}=[a_0-a_{1}x+a_2{x}^2-\cdots +a_{100}{x}^{100}]-[\cos (-\frac{\pi }{2}x)+\cos (-\frac{\pi }{2}x+\frac{\pi }{2})+\cdots +\cos (-\frac{\pi }{2}x+\frac{100\pi }{2})]\cdots (2)$

Adding $\left(1\right)$ and $\left(2\right),$ we get
$2(1+x^3{)}^{100}+2(1-x^3{)}^{100}=2(a_0+a_2{x}^2+\cdots +a_{100}{x}^{100})$
Putting $x=1,$ we get
$a_0+a_2+a_4+\cdots +a_{100}=2^{100}$
$\therefore k=100$