BYTETOOLS

Nth Term Calculator

Paste a sequence and find the rule behind it. Detects arithmetic, quadratic, cubic and geometric patterns and predicts any term you ask for.

Quadratic
Pattern
5
Terms given
101
Term 10

Rule found

  • aₙ = n² + 1

    Quadratic. Difference row 2 is constant at 2, so the rule is a degree-2 polynomial.

a10 = 101

Finite-difference table

Successive differences between the terms of the sequence
terms25101726
Δ3579
Δ2222

Row Δ2 is constant, which means the sequence is a degree-2 polynomial in n.

What is the Nth Term Calculator?

The ByteTools Nth Term Calculator finds the formula behind a sequence you paste in. It builds the finite-difference table, and when a difference row turns out constant it fits the matching polynomial rule — linear, quadratic aₙ = an² + bn + c, cubic or quartic — by solving the small linear system from the first terms.

  • Finite-difference table built and displayed in full
  • Fits linear, quadratic, cubic and quartic rules automatically
  • Detects geometric sequences by constant ratio
  • Detects two-term recurrences such as Fibonacci-style sequences
  • Predicts any term once a rule is found
  • Says honestly when no rule fits, instead of guessing

How to use the Nth Term Calculator

  1. 1

    Paste your sequence into the box, separating terms with commas, spaces or new lines.

  2. 2

    Enter the term number you want to predict.

  3. 3

    Read the detected pattern and the formula for aₙ.

  4. 4

    Check the finite-difference table to see which row became constant.

  5. 5

    Click Copy rule to save the formula and the predicted term.

About the Nth Term Calculator

The ByteTools Nth Term Calculator finds the formula behind a sequence you paste in. It builds the finite-difference table, and when a difference row turns out constant it fits the matching polynomial rule — linear, quadratic aₙ = an² + bn + c, cubic or quartic — by solving the small linear system from the first terms.

It also tests for a constant ratio, which gives the geometric rule aₙ = a₁rⁿ⁻¹, and for two-term recurrences where each term is built from the two before it, as in Fibonacci-style sequences. Once a rule is found you can predict any term you like.

The full difference table is shown so you can see why the answer is what it is. If no rule fits, the tool says so plainly instead of forcing one. Everything runs locally in your browser.

Frequently asked questions

How do you find the nth term of a sequence?

Take differences between consecutive terms. If those are constant the rule is linear; if the second differences are constant it is quadratic, and so on. This calculator builds that table and solves for the coefficients automatically.

How do you find the nth term of a quadratic sequence?

When the second differences are constant, the rule has the form an² + bn + c. Half the constant second difference gives a, and the remaining coefficients follow from the first few terms. The tool solves the whole system for you and shows the tidy formula.

What is the difference between an arithmetic and a geometric sequence?

An arithmetic sequence adds a fixed amount each time, so its first differences are constant. A geometric sequence multiplies by a fixed ratio each time. This calculator checks for both and tells you which pattern it found.

Why can't the calculator find a rule for my sequence?

It recognises polynomial sequences up to degree four, geometric sequences and two-term recurrences. Sequences built another way — prime numbers, digits of π, look-and-say — genuinely have no such formula, and the tool says so rather than fitting a false one.

How many terms do I need to enter?

At least three, but more is better. A quadratic rule needs three terms to pin down and a cubic needs four; extra terms let the calculator verify the rule against them rather than just fitting the minimum.

Related tools