Am sorry if i might seem a little noob, but my lecturer taught us to prove a few proof of functions by induction and even gave a few examples e.g
$1 + 3 + 5 + 7 + ... + 2n-1 = n^2$
But seriously he did'nt event tell us what proving a function by induction is. Please if someone would provide a brief and well understandable intro to proving functions by induction is. I'd appreciate! And to use as an example, is that equation there. Thanks in advance
Suppose you have... something. Like $1+3+5+\dots+(2n-1)$. You notice that
$$1=1^2\\1+3=2^2\\1+3+5=3^2$$
And you want to prove the general statement.
Then, induction goes like this:
Suppose it happens to be true for $k$. That is,
$$\underbrace{1+3+5+\dots}_k=k^2$$
Then, use this to prove it is true for $k+1$.
$$\begin{align}\underbrace{1+3+5+\dots}_{k+1}&=\underbrace{1+3+5+\dots}_k+(2k+1)\\&=k^2+2k+1\\&=(k+1)^2\end{align}$$
So, if your formula is true for $k=1$ (it is, we checked above), then it must be true for $k+1$, i.e. it is true for $k=2$.
If your formula is true for $k=2$, then it must also be true for $k+1$...
etc.