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.


If Bill Gates walked into a crowded bar, on an average, everyone is a millionaire. 

Loosely, an average is supposed to be a representative value for a sample. Sort of. But as you can see, it needn't be the case always. 
But every time, average is definitely this: if what we collectively have is distributed equally among all of us, the average is what each of us would get.
What we collectively have: $a_1+a_2+\cdots+a_n$
How many of us are there: $n$
What each one would get: $\frac{\text{total}}{\text{number of people}} = \frac{a_1+a_2+\cdots+a_n}{n} = \textbf{average}$
And that's why everyone becomes a millionaire. On average. Bill Gates simply has that much. The moral is, outliers can sometimes mess up the average, make it unreliable. Other times, everyone kind of has that much. 
PS: Call it arithmetic mean instead of average. Also, read the answer by @symplectomorphic. It has an interesting (and often very useful) take on how to think of an arithmetic mean. 
