Number Sequence Calculator
Arithmetic Sequence Calculator
definition: an = a1 + f × (n-1)
example: 1, 3, 5, 7, 9 11, 13, ...
Geometric Sequence Calculator
definition: an = a × rn-1
example: 1, 2, 4, 8, 16, 32, 64, 128, ...
Fibonacci Sequence Calculator
definition: a0=0; a1=1; an = an-1 + an-2;
example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
In math, a sequence is a list where each element follows a specific order. When the elements are numbers, we talk about a number sequence—a ordered collection that obeys a certain rule. Each element is called a term, and the total count of terms is its length, which may be infinite. The arrangement matters, and a term can appear more than once depending on the pattern. Several families of number sequences exist; the most frequently encountered are arithmetic, geometric, and Fibonacci sequences.
Sequences play a crucial role across many branches of mathematics because of their convergence behavior. A series is said to converge when its underlying sequence approaches a finite limit; otherwise the series diverges. Researchers use sequences to explore functions, spaces, and other structures. They serve as the foundation for series—essentially the summation of infinitely many values—widely applied in differential equations and analysis. Sequences can be presented by simply listing their elements when the pattern is obvious, or by using an indexing notation that provides a formula for the nth term as a function of n.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms stays constant. This constant gap may be positive or negative; its sign determines whether the terms head toward positive or negative infinity. The standard expression for an arithmetic sequence can be written as:
an = a1 + f × (n-1) or more generally | where an refers to the nth term in the sequence | |
| an = am + f × (n-m) | a1 is the first term | |
| i.e. | a1, a1 + f, a1 + 2f, ... | f is the common difference |
| EX: | 1, 3, 5, 7, 9, 11, 13, ... | |
It is clear in the sequence above that the common difference f, is 2. Using the equation above to calculate the 5th term:
| EX: | a5 = a1 + f × (n-1) a5 = 1 + 2 × (5-1) a5 = 1 + 8 = 9 | |
If we revisit the example, the fifth term, a5, computed with the given formula matches the displayed sequence, confirming correctness. Frequently, one also wants to find the total of an arithmetic series, which can be obtained by applying the following sum formula together with the earlier expression for an:
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Using the same number sequence in the previous example, find the sum of the arithmetic sequence through the 5th term:
| EX: |
1 + 3 + 5 + 7 + 9 = 25 (5 × (1 + 9))/2 = 50/2 = 25 | |
Geometric Sequence
A geometric sequence is a number sequence in which each successive number after the first number is the multiplication of the previous number with a fixed, non-zero number (common ratio). The general form of a geometric sequence can be written as:
| an = a × rn-1 | where an refers to the nth term in the sequence | |
| i.e. | a, ar, ar2, ar3, ... | a is the scale factor and r is the common ratio |
| EX: | 1, 2, 4, 8, 16, 32, 64, 128, ... | |
In the example above, the common ratio r is 2, and the scale factor a is 1. Using the equation above, calculate the 8th term:
| EX: |
a8 = a × r8-1 a8 = 1 × 27 = 128 | |
When we calculate the result with the formula and compare it to the earlier geometric sequence, the values coincide. The formula for the sum of a geometric sequence is:
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Using the same geometric sequence above, find the sum of the geometric sequence through the 3rd term.
EX: 1 + 2 + 4 = 7
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= |
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= 7 |
Fibonacci Sequence
A Fibonacci sequence generates each term by adding the two preceding numbers. Depending on the chosen start, the first two terms are either 1 and 1 or 0 and 1. Fibonacci numbers appear repeatedly—and sometimes surprisingly—in mathematics and have been extensively studied. They appear in algorithms (for example, Euclid’s method for the greatest common factor), in economics, and in natural phenomena such as tree branching or the bloom of an artichoke. In formula form, the Fibonacci sequence is expressed as:
| an = an-1 + an-2 | where an refers to the nth term in the sequence | |
| EX: | 0, 1, 1, 2, 3, 5, 8, 13, 21, ... | a0 = 0; a1 = 1 |