What is the intuition behind the formula for the average? Why is the average for $n$ numbers given by $(a+b+c+\cdots)/n$? I deduced the formula for the average of 2 numbers which was easy because its also the mid point, but I couldn't do it for more than 2 numbers.

In most contexts, what passes for an 'average' can be thought of this way: if you replaced a collection of separate instances with their 'average', you get the same result.
The usual mean comes from thinking this way for addition: if you have numbers $a_1,\ldots,a_n$, their sum is $a_1+\cdots+a_n$.  If you replaced all of them with their mean $\mu$, you should also get $a_1+\cdots+a_n$. 
Therefore $\mu$ must satisfy
$$
n\mu=a_1+\cdots+a_n,
$$
leading to the formula you've seen.
As another example: doing the same thing but for multiplication leads to the geometric mean $\sqrt[n]{a_1a_2\cdots a_n}$.
