The above proof is in my textbook, and in the last line it says $10*10^{k+1} = 10^{k+1}$. I just wanted to be sure that I'm not missing anything: as far as I can tell, this is a typo (the right hand side should've been $10^{k+2}$, I think)
Can anyone confirm that it is a typo?
Yes, it is a typo. At any rate, it is $10^{k+2}$ which is the end goal for the proof of the induction step.