Q:

How to prove that exists perfect square between the number $n10^n$ and $(n+1)10^n$?

I tried with mathematics induction, but I couldn't prove.

A:

Hint: suppose not, then there is a largest $m$ with $m^2\lt n\cdot10^n$ and we also have $(m+1)^2\gt (n+1)\cdot 10^n$
What can you then say about $(m+1)^2-m^2$?
And take it from there to get a contradiction.

