Adding It All Up

Need the total of every term, not just one? There is a shortcut. Pair the first term with the last, the second with the second-to-last, and so on. In an arithmetic sequence every pair adds to the same total, so the sum is just that total times the number of pairs.

That gives the series formula:

Sum = n/2 × (first + last)

Take 2, 4, 6, 8, 10. First plus last is 12, and there are 5 terms: 5/2 × 12 = 30. One multiplication replaces all the adding.

That sum formula quietly runs through daily life. A loan repaid in steadily rising installments, the total distance covered when speed climbs by a fixed amount each second, the number of times a countdown loop runs in code — all are arithmetic series.

The lesson is the same every time: when equal increments pile up, you never have to add them one by one. Reach for the pairing trick and its formula, and the running total drops out instantly.