What is 1 plus 2 plus 3 plus 4 and so on up to 20?

Answer 1

The easiest way to think of it is like so:

1 + 2 + 3 + .... + 18 + 19 + 20
= (1 + 20) + (2 + 19) + (3 + 18) + ... + (9 + 12) + (11 + 10)
= 21 + 21 + 21 + ... + 21 + 21 + 21

Now consider how many times that pattern will repeat. Each time we take two of those numbers and add them together, we're pulling two of them out of the sequence. That means that the total number of times we add them will always be half the length of the sequence. In this case, the length of the sequence is 20, so we'll be adding them together 10 times. In other words, the sum of the sequence will be:

(21 * 20) / 2
= 210

More generically, the sum of the numbers in an integer sequence from 1 up to any value (call it x), will be:

(x + 1)(x / 2), or (x2 + x)/2