Arithmetic Sequence And Series

An arithmetic sequence changes by the same amount each step. That fixed change is the common difference. An arithmetic series is the sum of terms from an arithmetic sequence.

If the first term is $a_1$ and the common difference is $d$, the $n$th term is

$$a_n = a_1 + (n - 1)d$$

If you want the sum of the first $n$ terms, use

$$S_n = \frac{n}{2}(a_1 + a_n)$$

This sum formula applies when you are adding the first $n$ terms of an arithmetic sequence. If you do not already know the last term, you can first find $a_n$ with the term formula.

How To Recognize An Arithmetic Sequence

A sequence is arithmetic only if the difference between consecutive terms stays constant.

For example, $4, 7, 10, 13, 16$ is arithmetic because each term increases by $3$. That means the common difference is $d = 3$.

By contrast, $5, 9, 14, 20$ is not arithmetic because the differences are $4$, $5$, and $6$. Since the difference changes, the arithmetic formulas do not apply.

Arithmetic Sequence Vs. Arithmetic Series

This distinction matters because one question asks for a term and the other asks for a total.

An arithmetic sequence is the ordered list itself. An arithmetic series is the result of adding the terms in that list.

For $2, 5, 8, 11$, the sequence is $2, 5, 8, 11$. The corresponding series is

$$2 + 5 + 8 + 11 = 26$$

Worked Example: Find The $20$th Term And The First $20$-Term Sum

Consider the arithmetic sequence

$$5, 8, 11, 14, 17, \ldots$$

Here, $a_1 = 5$ and $d = 3$.

Find The $20$th Term

Use

$$a_n = a_1 + (n - 1)d$$

Substitute $n = 20$:

$$a_{20} = 5 + (20 - 1)(3)$$ $$a_{20} = 5 + 57 = 62$$

So the $20$th term is $62$.

Find The Sum Of The First $20$ Terms

Now use

$$S_n = \frac{n}{2}(a_1 + a_n)$$

with $n = 20$, $a_1 = 5$, and $a_{20} = 62$:

$$S_{20} = \frac{20}{2}(5 + 62)$$ $$S_{20} = 10 \cdot 67 = 670$$

So the sum of the first $20$ terms is $670$.

Why The Arithmetic Series Formula Works

The first and last terms have the same average as the second and second-to-last terms, and the same pattern continues inward. In an arithmetic sequence, those pairs always add to the same total.

That is why the sum can be written as

$$\text{number of terms} \times \text{average of first and last term}$$

which becomes

$$S_n = \frac{n}{2}(a_1 + a_n)$$

This idea works only when the terms come from an arithmetic sequence, so the constant difference condition matters.

Common Mistakes With Arithmetic Sequence And Series Formulas

Mixing Up $n$ And $d$

$n$ counts the position or the number of terms. $d$ is the constant difference. They do different jobs in the formulas.

Forgetting The $(n - 1)$

The term formula is

$$a_n = a_1 + (n - 1)d$$

not $a_1 + nd$. There are only $n - 1$ jumps from the first term to the $n$th term.

Using The Sum Formula On A Non-Arithmetic List

If the differences are not constant, do not use the arithmetic series formula. Check the pattern first.

Losing The Sign Of The Difference

If the sequence decreases, then $d$ is negative. For example, in $12, 9, 6, 3$, the common difference is $-3$, not $3$.

When Arithmetic Sequences And Series Are Used

Arithmetic sequences show up whenever a quantity changes by a constant amount each step. Common examples include saving the same amount each month, rows of seats that increase by a fixed number, and algebra problems built on linear growth.

They are useful when change is additive rather than multiplicative. If each step multiplies by the same factor instead of adding the same amount, you are dealing with a geometric sequence instead.

Try A Similar Problem

Use the sequence $18, 15, 12, 9, \ldots$ to find the common difference, the $12$th term, and the sum of the first $12$ terms.

If you want a useful follow-up, solve the same kind of problem for a geometric sequence and compare what changes when the pattern is constant multiplication instead of constant addition.