Math Tools

Number Sequence Calculator

Calculate the nth term, sum, and first terms of arithmetic, geometric, or Fibonacci sequences.

Sequence Type

First Term (a)

Common Difference (d)

Position (n)

nth Term

Sum (first 10 terms)155

First ~10 Terms2, 5, 8, 11, 14, 17, 20, 23, 26, 29

What is the Number Sequence Calculator?

A sequence is an ordered list of numbers following a specific pattern. Arithmetic sequences have a constant difference between consecutive terms (e.g., 2, 5, 8, 11, ...). Geometric sequences have a constant ratio (e.g., 3, 6, 12, 24, ...). Fibonacci sequences begin 0, 1 and each next term is the sum of the previous two (e.g., 0, 1, 1, 2, 3, 5, 8, ...).

How it works

For arithmetic sequences: provide the first term (a) and common difference (d). The nth term is a + (n−1)d; the sum of the first n terms is n/2 × (2a + (n−1)d). For geometric sequences: provide the first term (a) and common ratio (r). The nth term is a × r^(n−1); the sum is a × (1 − r^n) / (1 − r). For Fibonacci: provide only n, and the calculator generates the first n Fibonacci numbers.

Arithmetic: nth term = a + (n−1)d; Sum = n/2(2a + (n−1)d). Geometric: nth term = a × r^(n−1); Sum = a(1 − r^n) / (1 − r). Fibonacci: F(n) = F(n−1) + F(n−2).

Arithmetic uses a starting value (a) and a constant difference (d) to jump by fixed steps. Geometric uses a starting value and multiplies by a constant ratio (r) each step. Fibonacci starts with 0 and 1, and every next term is the sum of the two previous ones.

Examples

Input	Result	Notes
Arithmetic: a=2, d=3, n=10	nth term = 29, Sum = 155	Sequence: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. Sum of all 10 terms: 155.
Geometric: a=3, r=2, n=6	nth term = 96, Sum = 189	Sequence: 3, 6, 12, 24, 48, 96. Sum: 3(1−2^6)/(1−2) = 189.
Fibonacci: n=10	nth term = 34, Sum = 88	First 10 Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. Sum: 88.

How to use the Number Sequence Calculator

Choose your sequence type from the dropdown menu

Enter the first term (a) and common difference (d) for arithmetic, or first term (a) and common ratio (r) for geometric

For Fibonacci, only specify the position n

Enter the position n to find the nth term

Read the nth term, sum, and displayed sequence terms in the results

Benefits

Handles three major sequence types in one tool — arithmetic, geometric, and Fibonacci

Instant nth term calculation — no need to compute by hand or use separate formulas

Shows the sum and first ten terms — get a complete picture of the sequence

Works with decimals and negative numbers — flexible for any starting values

Free, fast, and always available — no login or cost

Tips & common mistakes

Common mistakes

Confusing common difference (arithmetic) with common ratio (geometric) — difference means subtract, ratio means divide/multiply

Using the wrong formula for the wrong sequence type — each type has its own nth term formula

Entering zero as the common ratio in a geometric sequence — this causes division by zero; use a non-zero ratio

Forgetting that Fibonacci starts at 0, not 1 — the first term is 0, the second is 1

Tips

Arithmetic sequences have a linear pattern; think of them as a starting point plus repeated steps

Geometric sequences grow or shrink exponentially; a ratio greater than 1 causes rapid growth, between 0 and 1 causes decay

Fibonacci sequences appear in nature (flower petals, spiral shells) and in financial analysis — the golden ratio emerges from Fibonacci

Use this tool to check homework or explore how changing the first term or ratio affects the sequence

Frequently asked questions

What's the difference between arithmetic and geometric sequences?

Arithmetic sequences add the same number each time (common difference d). Geometric sequences multiply by the same number each time (common ratio r). For example, 2, 4, 6, 8, ... is arithmetic (add 2 each time); 2, 4, 8, 16, ... is geometric (multiply by 2 each time).

Can I use negative numbers in sequences?

Yes. Arithmetic sequences can have a negative difference (decreasing sequence). Geometric sequences can have a negative ratio (signs alternate). Fibonacci only produces non-negative integers by definition, but you can explore variations.

What if my common ratio is a fraction?

Fractions work fine. For example, a ratio of 0.5 means each term is half the previous one. A ratio of 1/3 produces a rapidly decreasing sequence. Use decimals (0.5, 0.333) in the input.

How many terms can I calculate?

You can specify any position n. The calculator shows the nth term and the sum. It displays the first ten terms for reference; the actual sequence can extend much further.

Why does my geometric series sum show a large number?

If the ratio is greater than 1, the terms grow exponentially. The sum of a geometric sequence grows very quickly. If the ratio is between 0 and 1, the sum converges to a finite value.

What's the golden ratio connection to Fibonacci?

As n increases, the ratio of consecutive Fibonacci numbers (F(n+1) / F(n)) approaches the golden ratio, approximately 1.618. This ratio appears in art, architecture, and nature.