Mathematical Induction and Powers

Question

If $c>1$ Prove that $c^m>c^n$ if and only if $m>n$. I don't know how to apply mathematical induction to this statement. Can anyone help?

Answer 1

I'll assume c $\in \mathbb{Z}$ and $m,n >1$

Check if true for $c= 2$:

Note that $2^m, 2^n>0$

We have that: $2^m - 2^n> 0 \Leftrightarrow 2^{m} > 2^{n}\Leftrightarrow 2^{m-n} > 1 \Leftrightarrow m-n > 0 \Leftrightarrow m>n$

There it's true for $n=1$ as each statement was an if and only if one.

Now we assume true for $c$:

$c^{m} > c^{n} \Leftrightarrow m>n$

Then we must prove it's also true for $c+1$:

$(c+1)^{m} > (c+1)^{n} \Leftrightarrow (c+1)^{m-n} > 1 \Leftrightarrow m>n $

Hence true $\forall c \in \mathbb{Z}^{+}$

Note that we could have proved this without induction, using the proof we did for $c+1$.

Answer 2

$\dfrac{c^m}{c^n}=c^{m-n} \gt 1 \text{ for } m\gt n $

$\implies c^m \gt c^n \text{ for } m\gt n $