Math induction problem (rules of exponents)

Question

Hello I am doing some induction problems, I have to prove that $3^{k+1}-1$ is a multiple of 2.

Suddenly they make this statement; $3^{k+1}$ is also $3 * 3^k$. Why is that?

Answer 1

In order to understand why $3^{k+1}=3\cdot 3^k$, you first need to understand what $3^{m}$ is, where $m$ is a natural number.

By definition: $3^0=1$ and $3^{m+1}=3\cdot 3^m$, for all $m\in \Bbb N_0\color{grey}{ =\{0, 1, 2, \ldots\}}$.

The equality $3^{k+1}=3\cdot 3^k$ is a mere consequence of the definition above.

Some authors will write $3^m=\underbrace{3\cdot 3\cdot \ldots \cdot 3}_{m\text{ factors}}$, but that's hardly rigorous and such a thing should only be written if the intended reader knows about the definition I used or if the reader doesn't know enough mathematics to deal with the definition above.