Number Sequence Calculator
Calculate arithmetic, geometric, and Fibonacci sequences with this free online calculator.
Arithmetic Sequence Calculator
An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
\(a_n = a_1 + d \times (n-1)\)
Where:
\(a_1\) = first term
\(d\) = common difference
\(n\) = position in sequence
Example: 1, 3, 5, 7, 9, 11, 13, ...
First term (\(a_1\)) = 1
Common difference (\(d\)) = 2
Geometric Sequence Calculator
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
\(a_n = a_1 \times r^{(n-1)}\)
Where:
\(a_1\) = first term
\(r\) = common ratio
\(n\) = position in sequence
Example: 1, 2, 4, 8, 16, 32, 64, 128, ...
First term (\(a_1\)) = 1
Common ratio (\(r\)) = 2
Fibonacci Sequence Calculator
The Fibonacci sequence is a sequence where each number is the sum of the two preceding ones, starting from 0 and 1.
\(F_0 = 0\)
\(F_1 = 1\)
\(F_n = F_{n-1} + F_{n-2}\) for \(n > 1\)
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
About Number Sequences
Number sequences are ordered lists of numbers that follow specific patterns or rules. They are fundamental in mathematics and have numerous applications in various fields.
Types of Sequences
There are many types of number sequences, each with its own pattern and formula. The most common ones include:
Arithmetic Sequence
An arithmetic sequence has a constant difference between consecutive terms. The formula for the nth term is:
\(a_n = a_1 + d(n-1)\)
Where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the position in the sequence.
Geometric Sequence
A geometric sequence has a constant ratio between consecutive terms. The formula for the nth term is:
\(a_n = a_1 \times r^{(n-1)}\)
Where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position in the sequence.
Fibonacci Sequence
The Fibonacci sequence is a special sequence where each number is the sum of the two preceding ones. It starts with 0 and 1, and the formula for the nth term is:
\(F_n = F_{n-1} + F_{n-2}\)
Where \(F_0 = 0\) and \(F_1 = 1\).
Applications of Number Sequences
- Mathematics: Number sequences are fundamental in various areas of mathematics, including calculus, number theory, and combinatorics.
- Finance: Sequences are used in calculating compound interest, loan payments, and investment growth.
- Computer Science: Sequences are used in algorithm analysis, data structures, and cryptography.
- Nature: The Fibonacci sequence appears in various natural phenomena, such as the arrangement of leaves on a stem, the spiral of shells, and the branching of trees.
- Art and Architecture: Sequences, especially the Fibonacci sequence and the golden ratio, are used in design, composition, and aesthetics.