Putting $x=1$ and $x=-1$ and adding, we get,

 $2\left(a_0+a_2+a_4+.\dots \dots ..+a_{50}\right)=3^{25}+1={\left(1+2\right)}^{25}+1$
$={{}_{}{}^{25}C}_0+{{}_{}{}^{25}C}_1\cdot 2+{{}_{}{}^{25}C}_2\cdot 2^2+.\dots \dots +{{}_{}{}^{25}C}_{25}\cdot 2^{25}+1$
$=2\left[1+{{}_{}{}^{25}C}_1+{{}_{}{}^{25}C}_2\cdot 2+.\dots \dots +{{}_{}{}^{25}C}_{25}\cdot 2^{24}\right]$
$\Rightarrow a_0+a_2+a_4+.\dots \dots \dots +a_{50}=1+25+{}_{}{}^{25}C_2\cdot 2+.\dots \dots +{}_{}{}^{25}C_{25}\cdot 2^{24}$
$=2\left[13+{{}_{}{}^{25}C}_2+{{}_{}{}^{25}C}_3\cdot 2+.\dots \dots \dots +{{}_{}{}^{25}C}_{25}\cdot 2^{23}\right]$ which is an even number.