How does sum of first $k$ integers equal $k(k+1)/2$
Let me tell you the story of young Carl Friedrich Gauss. 

He was six years old and in a small school with one class for everyone from 6 to 16. His teacher needed some quiet time for some job, so he asked the kids to add up the numbers 1 to 1000.

30 seconds later young Carl Friedrich had the answer: 500,500. 

How did he do it?

He changed the order and added 1+1000, 2+999, 3+998 up to 500+501. Each sum is 1001, and there are 500 sums, so the result is 1001 * 500. Obviously k=1000, so this is (k+1) * k / 2. 

You can use this for adding numbers in any arithmetic progression: sum = (first + last) * count / 2.