Understanding Sequences
In mathematics, a sequence is an ordered list of objects or numbers that follow a particular pattern. The individual elements in a sequence are called terms. The length of a sequence can be finite or infinite.
Sequences are foundational to understanding series (the sum of sequences) which are extensively used in calculus and mathematical analysis. Three of the most common number sequences are Arithmetic, Geometric, and Fibonacci.
Sequence Types
- ArithmeticAdd a constant difference
- GeometricMultiply by a constant ratio
- FibonacciSum of previous two terms
Arithmetic Sequence
An arithmetic sequence is a sequence in which the difference between each successive term remains constant. For example: 1, 3, 5, 7, 9 (difference is 2).
Where a₁ is the first term, and f is the common difference.
Geometric Sequence
A geometric sequence is a sequence in which each successive number is generated by multiplying the previous number by a fixed, non-zero number. For example: 1, 2, 4, 8, 16 (ratio is 2).
Where a₁ is the first term, and r is the common ratio.
Fibonacci Sequence
A Fibonacci sequence is a sequence in which every number following the first two is the sum of the two preceding numbers. It appears unexpectedly often in nature, such as in the branching of trees or the arrangement of leaves on a stem.