Nth Term Calculator

How the difference table works

Write the terms in a row. Underneath, write the gaps between them. Underneath that, the gaps between the gaps. Keep going until a row is constant.

For 1, 4, 9, 16, 25 the first differences are 3, 5, 7, 9 — not constant. The second differences are 2, 2, 2 — constant. Two levels down means the rule is a degree-2 polynomial, a quadratic.

The level at which the table goes constant is the degree of the rule. Linear rules settle after one level, quadratics after two, cubics after three. A table that never settles is not a polynomial of low degree at all.

This is the method of finite differences, and it is the discrete cousin of differentiation: differencing a degree-n polynomial gives a degree-(n−1) one, exactly as differentiating does.

The four families

Arithmetic. Constant first difference. 4, 7, 10, 13 steps by 3 every time, so a_n= 4 + (n − 1)×3.

Geometric. Constant ratio. 2, 6, 18, 54 multiplies by 3 every time, so a_n = 2 × 3⁽ⁿ⁻¹⁾. Differences will never settle for these, which is why the ratios are checked separately.

Quadratic. Constant second difference. 3, 8, 15, 24, 35 has second differences of 2, so the rule is n² + 2n.

Fibonacci-like. Each term is the sum of the two before it. 2, 3, 5, 8, 13. There is no simple closed form here — the recursive rule IS the answer, and any closed form needs the golden ratio.

Why three terms prove nothing

Give me 2, 4, 6 and I will say a_n = 2n. I could equally say a_n= n³ − 6n² + 13n − 6, which also gives 2, 4, 6 and then continues 14 rather than 8.

Both are correct. Any finite list of terms is consistent with infinitely many rules, and for any next term you like there is a polynomial that produces it. Pattern-finding is not deduction; it is a bet on simplicity.

Which is a defensible bet, and worth making explicit rather than hiding. This page reports the simplest fit among the standard families and tells you how thin the evidence is. With five or more terms confirming a rule, the simplest fit is usually the intended one. With three, it is a guess with good manners.

So the confidence line is part of the answer, not a disclaimer attached to it.

Deriving a quadratic by hand

Worth knowing, because it is quick and the page shows the same table you would build.

Take 3, 8, 15, 24, 35. First differences: 5, 7, 9, 11. Second differences: 2, 2, 2. Constant, so the rule is an² + bn + c.

a is half the second difference, so a = 1. b comes from the first difference: the first one is 5, and the first difference of an² + bn + c at that point is 3a + b, so b = 5 − 3 = 2. c from the first term: 3 = 1 + 2 + c, so c = 0.

The rule is n² + 2n. Check it: n = 4 gives 16 + 8 = 24, which is the fourth term. Substituting back is the step worth not skipping.

When no rule is found

The page tests constant first differences, constant second differences, constant ratios and the Fibonacci rule. If none fits, it says so and stops.

That is deliberate. A formula fitted through any five points always exists — a degree-4 polynomial goes through any five — and reporting it would technically be correct and practically worthless. It would tell you nothing about the sequence and would predict the sixth term badly.

If you have a sequence with a real rule that this page does not find, the On-Line Encyclopedia of Integer Sequences is the right next stop: it holds hundreds of thousands of catalogued sequences and will identify most of them from a handful of terms.

Sequences with no simple formula

Plenty of important sequences have no closed form at all, and it is worth knowing that the absence is normal rather than a failure of technique.

The primes. 2, 3, 5, 7, 11, 13. No formula generates the nth prime. There are approximations — the nth prime is roughly n ln n — but nothing exact.

Collatz stopping times. 27 takes 111 steps while its neighbours take 10 and 18. No formula predicts it, and there may not be one.

Recursive definitions generally. The Fibonacci numbers have a closed form only because their recurrence is linear. Most recurrences do not, and the rule itself is the best description available.

Sources and methodology

The method of finite differences is standard, and the limits of pattern-fitting are worth citing too.

• The On-Line Encyclopedia of Integer Sequences — where a sequence with no obvious rule can be looked up directly
• National Council of Teachers of Mathematics — how the difference method is taught
• Arithmetic Sequence Calculator — for when you already know the rule and want the terms
• How we calculate — methodology

Method. The differences are computed as exact fractions, so 0.1, 0.2, 0.3 is recognised as arithmetic instead of failing on floating-point noise, and 1/2, 1/4, 1/8 is recognised as geometric. Four families are tested in order of simplicity and the first exact fit is reported — with the difference table shown as the working rather than as decoration. When none of them fits, the page says which tests were tried and returns no rule, because an invented formula would be worse than an admitted failure. That engine is verified on every change against 76 hand-written assertions, including that each of the six standard families is identified and continued with the correct next three terms, and that no rule is invented for lists that have none. The count and the per-case breakdown are published on the formula verification page.

Related calculators

Where this goes next:

More in Math, or browse all calculators.

Read the guide

Once you know the family, the Arithmetic and Geometric Sequence Calculators take it further — terms, partial sums, and for a geometric one whether the infinite series converges.