Calculators guide
How to Find the nth Term of a Sequence
Step-by-step methods for linear, quadratic and geometric sequences, how to tell which you have from the differences, and how to check whether a number is in the sequence.
By M2Toolkit Editorial TeamPublished 7 min read
Quick answer
Look at the differences between terms. If they're all the same number d, the nth term is dn + (first term − d). If the second differences are the same, it's quadratic: an² + bn + c with a = half the second difference. If each term is multiplied by the same number r, it's geometric: a₁ × rⁿ⁻¹.
For 3, 7, 11, 15… the difference is 4, so the nth term is 4n − 1. Paste any sequence into the Sequence Calculator to get the rule and the working.
The “nth term” is a formula that gives any term from its position: put in n = 1 and you get the first term, n = 100 and you get the hundredth, without listing the 98 in between. Almost every sequence you'll meet in school maths and aptitude tests follows one of four patterns, and differences between terms tell you which.
Linear (arithmetic) sequences
The terms go up or down by the same amount each time. Take 5, 8, 11, 14, 17.
- 1Find the difference: 8 − 5 = 3, and it stays 3. That's the number in front of n: 3n.
- 2Compare 3n with the sequence. 3n gives 3, 6, 9, 12… — each is 2 less than the real term.
- 3Add the gap: nth term = 3n + 2. Check: n = 4 → 14 ✓.
A quicker way to get the constant: subtract the difference from the first term. 5 − 3 = 2. That's the “zeroth term” — the number that would come before the first.
Decreasing sequences work the same way with a negative difference: 20, 16, 12, 8 has d = −4 and nth term 24 − 4n.
Quadratic sequences
If the gaps themselves grow by the same amount, the sequence is quadratic. This is the method GCSE mark schemes expect. Take 2, 7, 14, 23, 34:
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Term | 2 | 7 | 14 | 23 | 34 |
| 1st difference | 5 | 7 | 9 | 11 | |
| 2nd difference | 2 | 2 | 2 |
- 1The second difference is 2. Halve it to get the n² coefficient: 1n².
- 2Subtract n² from each term: 2 − 1, 7 − 4, 14 − 9, 23 − 16, 34 − 25 = 1, 3, 5, 7, 9.
- 3That leftover is linear: difference 2, so 2n − 1.
- 4Put them together: nth term = n² + 2n − 1. Check n = 3: 9 + 6 − 1 = 14 ✓.
If the third differences are constant instead, the sequence is cubic (the n³ coefficient is the third difference ÷ 6). 1, 8, 27, 64 is simply n³.
Geometric sequences
Each term is the previous one multiplied by the same number — the common ratio r. Divide any term by the one before: for 3, 6, 12, 24, r = 2.
Ratios can be fractions (64, 32, 16, 8 has r = ½) or negative (2, −6, 18, −54 has r = −3, so the signs alternate). Geometric sequences model compound growth — savings, populations, depreciation — which is why they grow so much faster than linear ones. The same idea powers the Compound Interest Calculator.
How to tell which type you have
| Check | If it's constant… | Example |
|---|---|---|
| 1st differences | Linear: dn + c | 4, 9, 14, 19 |
| Ratios (term ÷ previous) | Geometric: a·rⁿ⁻¹ | 5, 15, 45, 135 |
| 2nd differences | Quadratic: an² + bn + c | 3, 6, 11, 18 |
| Each term = sum of previous two | Fibonacci-type | 2, 3, 5, 8, 13 |
| Each term = previous × p + q | Recursive “multiply then add” | 2, 5, 11, 23 (×2 + 1) |
Is a number in the sequence?
Set the nth term equal to the number and solve for n. If n comes out as a positive whole number, it's in the sequence. Is 399 in 4n − 1? 4n − 1 = 399 → n = 100, so yes, it's the 100th term. Is 400? n = 100.25, so no.
For quadratics you solve a quadratic equation — the Quadratic Equation Solver helps — and keep only positive whole-number solutions.
Try the free Sequence CalculatorType the terms, then use “Is a number in the sequence?” to find its position, or jump to any term number.Traps to watch for
- Writing only the difference. For 3, 7, 11 the answer isn't 4n but 4n − 1 — always check n = 1.
- Too few terms. Any three numbers fit some quadratic. 1, 2, 4 could be doubling (8 next) or adding 1, 2, 3 (7 next). Ask for, or look for, a fourth term.
- Sequence vs series. The nth term is one term; the sum of the first n terms is a different formula. For an arithmetic sequence it's n(first + last) ÷ 2; for anything else use the Series Calculator.
- Fibonacci numbers. 1, 1, 2, 3, 5, 8 has no simple polynomial nth term; it's defined by adding the previous two. The Fibonacci Calculator gives any term exactly.
Frequently asked questions
What does the n stand for in the nth term?
The position of the term in the sequence: n = 1 for the first term, n = 2 for the second, and so on. The formula turns a position into the value at that position.
How do you find the nth term of a decreasing sequence?
Exactly the same way — the difference is just negative. 30, 26, 22, 18 has d = −4, so the nth term is −4n + 34, usually written 34 − 4n.
What if the differences aren't constant at any level?
Check the ratios for a geometric sequence, then look for a term-to-term rule such as adding the previous two terms. Some puzzle sequences follow patterns no formula captures — like the digits of π.