It looks like the first number was determined by adding 1 (to zero), the second number was determined by adding 2 to that, the third by adding an additional 3, etc.

So to find the nth term, you need to add 1 + 2 + 3 +... + n.

Luckily there's a shortcut for working out this sum.

1 + 2 + 3 +... + n = (n/2)(n+1)

If you want to know why, pretend you're trying to add the numbers 1 through 10.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =??

Try rearranging the order:

1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 =??

Notice that each set of two numbers adds to 11:

(1 + 10) + (2 + 9) + (3 + 8) + (4 + 7) + (5 + 6) =??

11 + 11 + 11 + 11 + 11 = 11 * 5 = 55

So the ten numbers could be divided into 5 pairs (hence the n/2 in the above shortcut), each of which adds up to 11 (hence the n+1 in the shortcut). Likewise, if you wanted to add up the numbers from 1 to 100, you'd have 50 pairs of numbers adding up to 101. The (n/2)(n+1) shortcut works for any n, even if it's odd.