Arithmetic Series

The sum of an arithmetic series

The formula is

Where:
a is first term
d is common difference
n is the number of terms
S_n is the sum of the first n terms

Worked Example 1

Find the sum of the first 22 terms of the arithmetic series 6 + 4 + 2 + .....

number of terms (n) = 22, first term (a) = 6, common difference (d) = -2

S_n = 11 x ((2 x 6) + (21 x -2)) = 11 x (12 + -42) = 11 x -30 = -330

Further Information

You may have spotted that the contents of the square brackets is first term 'a' plus last term 'a+(n-1)d'. A way to help visualise what is happening in the formula, is to write an arithmetic series down twice. Once forwards and once backwards. Note that each pair of numbers adds up to the same amount. (10 in the 1 to 9 series below.)

1	2	3	4	5	6	7	8	9
9	8	7	6	5	4	3	2	1