What is a Number Sequence?
A number sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. Sequences can be finite (with a limited number of terms) or infinite (continuing without end).
Common types of sequences include arithmetic sequences (where the difference between consecutive terms is constant), geometric sequences (where the ratio between consecutive terms is constant), and Fibonacci sequences (where each term is the sum of the two preceding terms).
Sequence Formulas
Here are the key formulas for different types of sequences:
Arithmetic Sequence
An arithmetic sequence has a constant difference between consecutive terms:
$a_n = a_1 + (n-1)d$
Where:
- $a_n$ is the nth term
- $a_1$ is the first term
- $d$ is the common difference
- $n$ is the position in the sequence
The sum of the first n terms of an arithmetic sequence is:
$S_n = \frac{n}{2}[2a_1 + (n-1)d]$ or $S_n = \frac{n}{2}(a_1 + a_n)$
Geometric Sequence
A geometric sequence has a constant ratio between consecutive terms:
$a_n = a_1 \cdot r^{n-1}$
Where:
- $a_n$ is the nth term
- $a_1$ is the first term
- $r$ is the common ratio
- $n$ is the position in the sequence
The sum of the first n terms of a geometric sequence is:
$S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$
$S_n = n \cdot a_1$ for $r = 1$
Fibonacci Sequence
In a Fibonacci sequence, each term is the sum of the two preceding terms:
$F_n = F_{n-1} + F_{n-2}$
With starting values typically $F_1 = 0$ and $F_2 = 1$
How to Use This Calculator
- Select the type of sequence you want to generate (arithmetic, geometric, or Fibonacci).
- Enter the required parameters for your chosen sequence type.
- Specify the number of terms you want to generate (maximum 100).
- Click 'Calculate' to generate the sequence and see the step-by-step solution.
Use arithmetic sequences for constant differences, geometric sequences for constant ratios, and Fibonacci mode when each term depends on the two previous values.
