The nth term of an arithmetic sequence is found using the formula

an=a1+(n−1)da_n=a_1+(n-1)dan​=a1​+(n−1)d

where a1a_1a1​ is the first term, ddd is the common difference, and nnn is the term number.

How to Find the nth Term of an Arithmetic Sequence

(Quick Scoop)

1. What an arithmetic sequence is

An arithmetic sequence is a list of numbers where you add the same number each time to get from one term to the next.

That repeated “jump” is called the common difference.

Example:

  • Sequence: 3, 7, 11, 15, 19, …
  • Each time you add 4, so the common difference d=4d=4d=4.

2. The key formula for the nth term

For any arithmetic sequence:

an=a1+(n−1)da_n=a_1+(n-1)dan​=a1​+(n−1)d

  • ana_nan​: the nth term (the term you want).
  • a1a_1a1​: the first term of the sequence.
  • ddd: the common difference between terms.
  • nnn: the position number in the sequence (1st, 2nd, 3rd, …).

This same formula appears in school resources and worked examples.

3. Step‑by‑step method (with examples)

Step 1: Identify the first term and the common difference

Take the sequence:
5, 8, 11, 14, 17, …

  • First term: a1=5a_1=5a1​=5.
  • Common difference: subtract any term from the next one:
    • 8−5=38-5=38−5=3
    • 11−8=311-8=311−8=3
      So d=3d=3d=3.

If you don’t know consecutive terms, you can still find ddd by dividing the change in value by the change in term number.

Step 2: Write the nth‑term formula for this sequence

Use the general formula an=a1+(n−1)d ,a_n=a_1+(n-1)d,an​=a1​+(n−1)d.

For this sequence:

an=5+(n−1)⋅3a_n=5+(n-1)\cdot 3an​=5+(n−1)⋅3

You can expand it if you like:

an=5+3n−3=3n+2a_n=5+3n-3=3n+2an​=5+3n−3=3n+2

So the nth term rule is:

an=3n+2a_n=3n+2an​=3n+2

Many school guides show rules like “3n + 2” or “4n − 1” as nth‑term formulas.

Step 3: Use the formula to find any term

Using an=3n+2a_n=3n+2an​=3n+2:

  • 1st term: a1=3(1)+2=5a_1=3(1)+2=5a1​=3(1)+2=5
  • 4th term: a4=3(4)+2=14a_4=3(4)+2=14a4​=3(4)+2=14
  • 10th term: a10=3(10)+2=32a_{10}=3(10)+2=32a10​=3(10)+2=32

You can jump to any position in the sequence just by plugging in nnn.

4. Example: direct use of the formula

Example 1: Find the 25th term

Sequence: 3, 9, 15, 21, 27, …
From a worked example:

  • a1=3a_1=3a1​=3
  • d=9−3=6d=9-3=6d=9−3=6
  • n=25n=25n=25

Use the formula:

a25=a1+(n−1)d=3+(25−1)⋅6a_{25}=a_1+(n-1)d=3+(25-1)\cdot 6a25​=a1​+(n−1)d=3+(25−1)⋅6

a25=3+24⋅6=3+144=147a_{25}=3+24\cdot 6=3+144=147a25​=3+24⋅6=3+144=147

So the 25th term is 147.

Example 2: When you know two non‑consecutive terms

Sometimes you are given, say, the 3rd term and the 5th term and asked to find the nth term.

Suppose:

  • a3=9a_3=9a3​=9
  • a5=15a_5=15a5​=15

We know:

a3=a1+2d=9a_3=a_1+2d=9a3​=a1​+2d=9

a5=a1+4d=15a_5=a_1+4d=15a5​=a1​+4d=15

Subtract the first equation from the second:

(a1+4d)−(a1+2d)=15−9(a_1+4d)-(a_1+2d)=15-9(a1​+4d)−(a1​+2d)=15−9

2d=6⇒d=32d=6\Rightarrow d=32d=6⇒d=3

Then plug back:

a3=a1+2⋅3=9⇒a1=3a_3=a_1+2\cdot 3=9\Rightarrow a_1=3a3​=a1​+2⋅3=9⇒a1​=3

So:

an=3+(n−1)⋅3=3na_n=3+(n-1)\cdot 3=3nan​=3+(n−1)⋅3=3n

This style of reasoning is also shown in worked arithmetic‑sequence problems.

5. Quick checklist and FAQ‑style tips

  • Always start at n=1n=1n=1 for the first term in standard school problems.
  • The common difference ddd can be positive, negative, or even decimal.
  • If the pattern doesn’t add or subtract the same amount each time, it’s not an arithmetic sequence, so this formula won’t work.
  • To check your rule, plug in n=1,2,3n=1,2,3n=1,2,3 and see if you recreate the original sequence.

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