Proof $\forall n\in\mathbb{N}$, that $9|10^n-1$ by mathematical induction

Solution 1:

I think you're almost all the way there-- the main feedback I'd give is around your final statement:

$$9|10^{n+1}-1\implies9|(9k+1)\cdot10-1\implies9|90k+9$$

Above you're assuming what you want to prove. Try working backwards starting with what you know to get to what you want to prove. Here, try the following to get started:

$$10^n = 9k + 1 \implies 10^{n+1} = (9k + 1) \cdot 10 \implies 10^{n+1} - 1 = ... $$

N.B. Also be sure to declare what $k$ is when you use it.

I hope this helps!

Solution 2:

When you write $9\mid10^n-1\implies9k=10^n-1$, you should say what that $k$ is. Something like “$9k=10^n-1$” for some non-negative integer $k$.

And then your aim is to prove that $9\mid10^{n+1}-1$. That's not what you do. You start from $9\mid10^{n+1}-1$ and then you see what you can conclude from that.