Still remember how to calculate the summation for 1+2+3+ … + 10 ? Well, the summation for the first ten sequence (starting from 1) is 55.

But, how do you calculate it? Pressing the sequence into calculator, one by one manually?

Fine. What if I change the question to 1+2+ …+ 1000. Probably, you will not bother to calculate this anymore.

Try to recall what you have learned in secondary school — there is a formula!

1+2+3+ …+ n = n × (n + 1) /2

Perhaps, the math teacher had challenged you to add from 1 to 100, or to any suggested number and he could give you the answer in 5 seconds. Later, he disclosed the secret formula to you and the whole class laughing together and learned the secret formula.

So, why 1+2+3+ …+ n = n × (n + 1) /2 ?

Perhaps your math teacher just ask you memorize this secret formula (that why you have forgotten?). In fact, there are a few proofs to derive this formula. In the following, I will provide a simple example to derive the summation formula. Be noted that, this is just a simple illustration and it is NOT a complete proof.

Okay. Assuming we have 4 x 4 = 16 balls that are arranged as the following picture.

This balls topology can further be segmented to two triangles one rectangle.