Putting $x=1$, we get,  

$3^{25}=a_0+a_1+a_2+\mathrm{.....}+a_{50}$    ....(1)

Putting $x=-1$, we get,    

$1=a_0-a_1+a_2+\mathrm{........}+a_{50}$    ...(2)

adding equation (1) & (2), we get,

$\Rightarrow a_0+a_2+\dots \dots +a_{50}=\frac{3^{25}+1}{2}=\frac{{\left(1+2\right)}^{25}+1}{2}$

$\Rightarrow a_0+a_2+a_4+\mathrm{.....}+a_{50}=\frac{\text{}\left({}^{25}{\text{C}}_0+{\text{}}^{25}{\text{C}}_1·2+ {\text{}}^{25}{\text{C}}_2·2^2+\dots + {\text{}}^{25}{\text{C}}_{25}·2^{25}\right)+1}{2}=\frac{2\left[1+{\text{}}^{25}{\text{C}}_1+ {\text{}}^{25}{\text{C}}_2·2+\dots + {\text{}}^{25}{\text{C}}_{25}·2^{24}\right]}{2}$

$\Rightarrow a_0+a_2+a_4+\mathrm{......}+a_{50}$$=2\left[13+{}^{25}{C}_2+\mathrm{......}+{}^{25}{C}_{25}{.2}^{23}\right]$

$a_0+a_2+a_4+\mathrm{......}+a_{50}=2K$$\Rightarrow$ even