Arithmetic sequences and series

Sequences with a constant common difference — the nth-term rule and the sum formula, and where each comes from.

Pure · Sequences & series

Arithmetic sequences and series

Sequences with a constant common difference — the nth-term rule and the sum formula, and where each comes from.

Why it works

An arithmetic sequence goes up (or down) by the same amount each step — the common difference

d

. Starting from the first term

a

a,\ a + d,\ a + 2d,\ a + 3d,\ \dots

By the

n

th term you've added

d

a total of

(n - 1)

times (not

n

— the first term has had no steps added), so

u_n = a + (n - 1)d.

The sum of the first

n

terms,

S_n

, has a neat shortcut. Write the sum forwards and backwards and add them in pairs: every pair totals

a + l

(first plus last), and there are

n

of them, so

2S_n = n(a + l)

. Hence

S_n = \frac{n}{2}(a + l) = \frac{n}{2}\big[2a + (n - 1)d\big],

using

l = a + (n-1)d

for the last term. The two forms are the same — use whichever fits what you're given.

The usual mistakes are arithmetic-vs-geometric (here you add

d

, you don't multiply) and the "

n - 1

" slip in the

n

th term.