Definition: An arithmetic series is the sum of the terms of an arithmetic sequence, whose consecutive terms differ by a constant amount called the common difference.
What is an arithmetic series?
Add the terms of a sequence that goes up (or down) by the same step d each time. The sum of the first n terms is
Sn = n(a1 + an)/2,
which is the number of terms times the average of the first and last term.
How do you use it?
Worked example: add 8 + 13 + 18 + … + 78. The common difference is 5. To find n, solve 8 + 5(n – 1) = 78, so n – 1 = 14 and n = 15.
Then S15 = 15(8 + 78)/2 = 15 times 43 = 645.
The pairing trick behind the formula: 8 + 78 = 86, 13 + 73 = 86, and so on, with the middle term 43 left over.
What is a common mistake?
Using n as the last term instead of the number of terms. Mathbot always finds n first by counting steps, then applies the formula.