An arithmetic sequence adds the same number at every step: 5, 9, 13, 17, and so on. If you know the first term and the step size, you can jump straight to any term without writing out the whole list. That shortcut is the nth-term formula.
How it works
Call the first term a, the common difference d, and the term number n. The nth term is a + (n – 1) x d. The (n – 1) matters: the first term needs zero steps to reach, the second needs one step, and the nth needs n – 1 steps. The common difference is any term minus the one before it, so check two pairs to be sure the sequence really is arithmetic.
Step by step
- Write down the first term a.
- Find the common difference d by subtracting the first term from the second, and confirm with another pair.
- Write down n, the term number you want.
- Use the formula: nth term = a + (n – 1) x d.
- Work out (n – 1) x d first, then add a.
- Check your answer by counting a few steps by hand for small values of n.
Worked example
Sequence 7, 13, 19, 25. First term a = 7, common difference d = 6. Find the 20th term: 7 + (20 – 1) x 6 = 7 + 114 = 121. To check, the 4th term should be 7 + 3 x 6 = 25, which matches the list. Another example with a negative difference: 40, 33, 26. Here d = -7. The 10th term is 40 + 9 x (-7) = 40 – 63 = -23.
Common mistakes
- Using n instead of n – 1, which adds one extra step and overshoots by d.
- Taking d from only one pair when the sequence is not arithmetic at all.
- Mixing up the sign of a negative common difference.
- Counting terms from zero instead of one, which shifts every answer by one step.
Try it with the free tool
You can follow every step above by hand, but the free tool below does the same calculation instantly. Try your own numbers in it, then check one result by hand so you trust the method as well as the answer. Open the full tool here: open the tool.
Checklist before you finish
- Write down every input number first, with its unit, before you calculate anything.
- Work through the steps in order, and do not skip the unit conversions.
- Check the answer against the worked example above: is it the right size, and does the direction make sense?
- Round only at the end, and keep the unrounded figure for any follow-up calculation.
Practice with your own numbers
Pick three cases of your own and work each one by hand. First, try a simple round-number case you can check mentally. Second, try a realistic case with awkward numbers, the kind you will actually meet. Third, try an edge case, such as a very small or very large input, and confirm the answer still behaves sensibly. If any case gives a surprising result, re-check the units and the formula step by step before you trust it.
If your answer looks wrong
Most errors come from a small set of causes. Check that you used the same units throughout, that you divided where the formula divides and multiplied where it multiplies, and that you applied any percentage as a decimal. Check the order of operations: powers and brackets come before multiplication and division, which come before addition and subtraction. Finally, test the method on the worked example in this guide. If your method reproduces the example answer, the method is right and the inputs deserve a second look.
Quick tips
If you are given a later term instead of the first, work backwards: subtract d the right number of times to reach the first term, then use the formula. For a term far out, like the 100th, the formula saves you from writing 100 numbers.