Sum of Indices of Real Number/Positive Integers

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Theorem

Let $r \in \R_{> 0}$ be a positive real number.

Let $n, m \in \Z_{\ge 0}$ be positive integers.

Let $r^n$ be defined as $r$ to the power of $n$.


Then:

$r^{n + m} = r^n \times r^m$


Proof

Proof by induction on $m$:

For all $m \in \Z_{\ge 0}$, let $\map P m$ be the proposition:

$\forall n \in \Z_{\ge 0}: r^{n + m} = r^n \times r^m$


$\map P 0$ is true, as this just says:

$r^{n + 0} = r^n = r^n \times 1 = r^n \times r^0$


Basis for the Induction

$\map P 1$ is true, by definition of power to an integer:

$r^{n + 1} = r^n \times r = r^n \times r^1$

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$\forall n \in \Z_{\ge 0}: r^{n + k} = r^n \times r^k$


Then we need to show:

$\forall n \in \Z_{\ge 0}: r^{n + k + 1} = r^n \times r^{k + 1}$


Induction Step

This is our induction step:


\(\displaystyle r^n \times r^{k + 1}\) \(=\) \(\displaystyle r^n \times \paren {r^k \times r}\) $\quad$ Definition of Integer Power $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \paren {r^n \times r^k} \times r\) $\quad$ Real Multiplication is Associative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle r^{n + k} \times r\) $\quad$ Induction Hypothesis $\quad$
\(\displaystyle \) \(=\) \(\displaystyle r^{n + k + 1}\) $\quad$ Definition of Integer Power $\quad$

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n, m \in \Z_{\ge 0}: r^{n + m} = r^n \times r^m$

$\blacksquare$


Sources