Arithmetic Sequences: The nth Term and Gauss's Sum Formula

Q: What is the formula for the nth term?

a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.

How to spot an arithmetic sequence, find any term with the $n$ th term formula, and add up a whole run of terms with the pairing trick attributed to the young Gauss.

3 min read Supports 3 lessons

A sequence is an ordered list of numbers, called terms, that follows a rule. In an arithmetic sequence the rule is the simplest possible: add the same number every time.

5,\; 8,\; 11,\; 14,\; 17,\; \dots \qquad d = 3

The common difference $d$ is the gap between consecutive terms.

a_{1}

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

Equal steps: the tops of the bars line up on a straight line.

The difference can be negative, as in $20, 17, 14, 11, \dots$ , where $d = -3$ and the terms go down.

Finding any term

To reach the $n$ th term from the first, you take $n - 1$ steps, each of size $d$ .

nth term

a_n = a_1 + (n - 1)d

$7,\; 11,\; 15,\; \dots \qquad a_{20} = \;?$
Find the 20th term.

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Watch out

It is $(n - 1)$ steps, not $n$ . The first term needs no steps at all.

Tip

Rearranged, $a_n = dn + (a_1 - d)$ is a straight line with slope $d$ . Arithmetic sequences are linear functions sampled at whole numbers.

Adding up the terms

Legend has it that a young Carl Friedrich Gauss was asked to add the numbers $1$ to $100$ and answered almost instantly. He paired the first with the last, the second with the second-last, and so on. Every pair made $101$ , and there were $50$ pairs.

$1 + 2 + 3 + \cdots + 98 + 99 + 100$

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Sum of the first n terms

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

Half the number of terms, times the first plus the last.

Worked example

Find the sum of the first $20$ terms of $7, 11, 15, \dots$

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From above, $a_{20} = 83$ .
$S_{20} = \frac{20}{2}(7 + 83)$

Key idea

This sum, written with sigma notation as $\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}$ , turns up again in calculus when areas are approximated by adding up thin rectangles.

Common questions

How do I know if a sequence is arithmetic?: Subtract each term from the next. If the difference is always the same, the sequence is arithmetic and that number is the common difference.
What is the formula for the nth term?: $a_n = a_1 + (n - 1)d$ , where $a_1$ is the first term and $d$ is the common difference.

Practise arithmetic sequences

The Arithmetic Sequences lesson has worked examples and unlimited practice that adapts to you.

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Common questions

How do I know if a sequence is arithmetic?

Subtract each term from the next. If the difference is always the same, the sequence is arithmetic and that number is the common difference.

What is the formula for the nth term?

$a_n = a_1 + (n - 1)d$ , where $a_1$ is the first term and $d$ is the common difference.