Number Sequence Calculator

Find missing numbers in a list, identify the sequence pattern type, and determine the general nth term formula.

Preparing Number Sequence Calculator

Please wait ...

Input

Enter your list of numbers separated by commas or spaces, choose the sequence type if known, and calculate the next terms in the series.

Input summary

Your calculator summary shows here.

Step by Step Calculation

↗

Step by step calculation shows here

Calculate first and the full working will appear below automatically.

A Guide to Solving Number Sequences and Series

A number sequence is simply an ordered list of numbers that follow a specific mathematical rule or pattern. Every number in that list is called a term. Finding the hidden rule in a sequence lets you predict what values will come next in the list and sum the values up. Analyzing sequences is a fundamental part of problem-solving, science, financial forecasting, and pattern recognition tests.

Main Types of Sequences

Most simple sequences belong to one of two primary categories. The first is an arithmetic sequence, where each term is found by adding or subtracting a constant value (called the common difference) to the previous term. The second is a geometric sequence, where each term is found by multiplying the previous term by a constant factor (called the common ratio).

Geometric sequences grow or shrink extremely quickly because they involve multiplying repeatedly. This multiplication relationship can also be expressed as raising values to powers, which you can evaluate using our raising numbers to powers tool. If you need to reverse power calculations to find scale levels, check out our logarithm calculations tool.

Real-World Patterns & Series

Financial Savings: If you deposit a fixed sum of money into a bank account every month, the balance forms an arithmetic sequence.
Population Growth: Animal populations or bacteria cultures grow by multiplying by a constant rate, forming a geometric sequence. You can track this alongside our percentage change tools.
Nature's Spirals: The famous Fibonacci sequence, where each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, 13...), models shell spirals and pinecones.
Building Steps: Stacking blocks or stairs creates steps that grow by a constant height difference, forming a simple arithmetic progression. You can calculate basic sums with our basic arithmetic operations tool.

The General nth Term Formula

To find a number deep in a sequence (like the 100th term) without writing out the whole list, mathematicians use a general nth term formula. For arithmetic lists, the formula is: term = first_term + (n - 1) × difference. For geometric lists, the formula is: term = first_term × ratio^(n - 1).

These formulas represent the mathematical rule of the pattern. If you want to check the middle value of a list of numbers, you can use our calculating group averages tool to evaluate the mean.

Example of an Arithmetic List

Suppose you have the list of numbers: 3, 7, 11, 15, 19...

By comparing the values, we see that each number increases by exactly 4. This tells us it is an arithmetic sequence with a first term of 3 and a common difference of 4. To find the 10th term, we apply the formula: 3 + (10 - 1) × 4 = 3 + 9 × 4 = 3 + 36 = 39. The 10th term in the sequence is exactly 39. This practical example illustrates how sequence rules save time and predict values.