Adding Lots of Numbers...The Fast Way

It is easy to notice that 1 + 100 = 101, 2 + 99 = 101, ... and so on.

This continues till 50 + 51 = 101.

$\therefore$ If we rearrange the addition we basically have 1 + 2 + 3 ... + 100 = 101 x 50 = $\Large\boxed{ 5050 }$

Alternatively ,

The formula for adding n natural numbers = $\Large\frac{ (n)(n+1) }{ 2 }$

$\therefore$ $\Large\frac{ (n)(n+1) }{ 2 }$ = $\Large\frac{ (100)(100+1) }{ 2 }$ = $\Large\frac{ 10100 }{ 2 }$ = $\Large\boxed{ 5050 }$

I'm sure there are many ways to do this problem, but the way I think of it is like this:

Picture every number in a row. 100 is an easy number to work with, we'll remove it for now, and add it back on later once we figure out the sum for 1 to 99. Speaking of the numbers 1 & 99, You might notice they add together to make 100. We'll remove them from the row. Continuing this train of thought, 2 & 98 make 100, as do 3 & 97, 4 & 96...In fact, we can pair every number on one side of the row to some other number on the other side of the row can't we? Well, we can, except for the number 50. There isn't a second 50 in our row of numbers, so whatever we get for adding up the rest of these numbers, we'll add 50 on the end.

Now, the exciting part. Every number from 1 to 49 can be added to a number from 51 to 99 to make 100. Effectively, meaning we can say that our answer is 49x100 plus all that other stuff we set aside for later.

Now if we add that stuff we set aside, we get

49x100 + 100 + 50.

Or, to simplify

50*100+50

or 5050, our final answer.