If $a_n=2^{2^n}+1$ then for  $n>1$ , last digit of $a_n$ is 

For $n=2,a_n=2^4+1=16+1=17$. . Assume 

That $a_k=2^{2^k}+1=10m+7$ where $k>1$

and m is some positive   integer.

Now, $\begin{array}{l}{a}_{k+1}=2^{2^{k+1}}+1={\left(2^{2^k}\right)}^2+1\\ =(10m+6{)}^2+1=10\left(10m^2+12m+3\right)+7\end{array}$

Thus, last digit of $a_n$  is $7\mathrm{\forall }n>1$