Sequence Calculator
Calculate arithmetic, geometric, Fibonacci, and custom sequences. Find nth terms, sums, and analyze convergence with step-by-step solutions.
Sequence Types & Formulas
A sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term.
Arithmetic Sequences
In an arithmetic sequence, each term differs from the previous by a constant value (common difference d).
Geometric Sequences
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio r.
Fibonacci Sequence
Each Fibonacci number is the sum of the two preceding ones. The ratio of consecutive terms approaches the golden ratio.
Convergence of Sequences
A sequence converges if its terms approach a specific value as n approaches infinity.
- $a_n = \frac{1}{n}$ → 0
- $a_n = \frac{n+1}{n}$ → 1
- $a_n = \frac{(-1)^n}{n}$ → 0
- $a_n = n$ → ∞
- $a_n = (-1)^n$ (oscillates)
- $a_n = 2^n$ → ∞
Common Sequence Types
| Type | Formula | Example |
|---|---|---|
| Natural Numbers | $a_n = n$ | 1, 2, 3, 4, ... |
| Perfect Squares | $a_n = n^2$ | 1, 4, 9, 16, ... |
| Powers of 2 | $a_n = 2^n$ | 2, 4, 8, 16, ... |
| Factorials | $a_n = n!$ | 1, 2, 6, 24, ... |
| Harmonic | $a_n = \frac{1}{n}$ | 1, 1/2, 1/3, 1/4, ... |
Sequence Tip:
When analyzing a sequence, look for patterns in differences between consecutive terms, ratios of consecutive terms, or try to express the nth term as a function of n.
Sequence Examples
Sequence Calculator Answers
What does the sequence calculator do?
The sequence calculator helps you find terms in a mathematical sequence, such as arithmetic or geometric sequences, based on the given formula or initial terms.
What types of sequences can it calculate?
It can compute arithmetic, geometric, harmonic, Fibonacci, and custom-defined sequences by identifying patterns and generating terms accordingly.
Can it find a specific term in the sequence?
Yes. You can input the position of the term you want, and the calculator will compute its value based on the rule or formula you provide.
Is it useful for students?
Definitely. It’s great for students learning about sequences and series in algebra or pre-calculus, helping them check their work and understand patterns.