An arithmetic sequence climbs (or falls) by the same amount at every step. That fixed gap is the common difference d. Starting from a first term a, you keep adding d:

This is exactly the linear nth-term rule seen from the front: the nth term is the first term plus (n-1) jumps of d.

A series is what you get when you add up the terms of a sequence. The sum of the first n terms of an arithmetic sequence is written S_n. We could just substitute and add one term at a time — but there is a far quicker way.

See it built

Here is the trick the young Gauss is said to have used to add 1 + 2 + \dots + 100 in seconds. Write the sum forwards, write it again backwards underneath, and add the two rows: every column makes the same total, (\text{first} + \text{last}). Step through it.

There are n columns, each adding to (\text{first} + \text{last}), and the two rows together count the sum twice. So:

Since the last term is a + (n-1)d, substituting gives the form you can use straight from a, d and n:

For 1 + 2 + \dots + 100 that is \tfrac{100}{2}(1 + 100) = 50 \times 101 = 5050.

See it explained

Sal Khan derives the arithmetic series formula by averaging the first and last terms.