8. The number $\left(10^{n}+10^{n-1}+\ldots+1\right) \cdot\left(10^{n+1}+5\right)+1$ is a perfect square.
8. $\left(10^{n}+10^{n-1}+\ldots+1\right)\left(10^{n+1}+5\right)+1=$ $=\frac{10^{n+1}-1}{10-1}\left(10^{n+1}+5\right)+1=\left(\frac{10^{n}+1+2}{3}\right)^{2}$. The number $\frac{10^{n}+1+2}{3}$ is an integer, since the number $10^{n+1}+2=1000 \ldots 02$ is divisible by 3, because the sum of its digits is divisible by 3.