Arithmetic Sequences and Series

Once the step between terms is constant, two formulas do all the work: one jumps straight to any term you want, and one adds up a whole stretch of them without touching a calculator more than a few times. A sequence is the list; a series is what you get when you add that list together.

Theorem The two core formulas

The th term is , and the sum of the first terms is .

Here is the first term, the common difference, and the position.

Term formula:

Sum, knowing both ends:

Sum, knowing instead:

Note the , not . Reaching the th term takes only eleven steps, because the first term is already there before any step is taken.

Concept Sigma notation

A series is often written compactly with the summation sign:

The letter starts at the value written below and climbs to the value written above, and is the rule generating each term. Everything the notation describes is still an ordinary series — the same two formulas apply.

Example Finding a distant term

a) Find the 12th term of

Common difference:

Substitute:

Simplify:

b) Write a rule for

Common difference:

Substitute:

Expand:

c) Given and , find

Substitute into the term formula:

Simplify:

Part (c) shows the formula works backwards just as happily. Any three of , , , determine the fourth.

Example Arithmetic means

Find four arithmetic means between and .

Four means plus the two end terms gives

The end terms are

and

Substitute:

Simplify:

, so

Build the sequence:

The four means are

The trap is counting . The means sit between the given numbers, so both ends must be counted too.

Example Summing a series

a) Find

Common difference:

Find how many terms:

Simplify:

, so

and

Apply the sum formula:

Simplify:

b) Evaluate

Number of terms:

First term at

Last term at

Apply the sum formula:

Simplify:

In both parts the work is the same: find , find the two end terms, then average them and multiply by how many there are. That is all the sum formula really says.

Summary

A sequence is a list of terms; a series is that list added together.
The th term is — note the , not .
Knowing both ends, the sum is .
Knowing instead, use .
Any three of , , , give you the fourth.
Inserting arithmetic means between two numbers makes terms in total.
Sigma notation is only shorthand — the same formulas evaluate it.