Sequence Calculator: Find Terms, Formulas, Sums and Patterns
Analyze number sequences, calculate arithmetic and geometric terms, find explicit formulas, generate recursive sequences, and view step-by-step solutions.
Enter existing terms and the calculator will check for common arithmetic, geometric, quadratic, or Fibonacci-like patterns.
Calculate the nth term, sequence sum, explicit formula, recursive formula, and generated terms.
Find the nth term, finite sum, possible infinite sum, explicit formula, recursive rule, and generated terms.
Enter sequence terms to identify a supported pattern and derive an explicit nth-term formula.
Generate recursively defined sequences using common recurrence rules without evaluating unsafe custom expressions.
Use the sequence calculator to analyze number patterns, find the nth term, calculate sequence sums, and work with arithmetic, geometric, explicit, and recursive formulas. You can enter an existing list of numbers or use known values such as the first term, common difference, common ratio, or term position.
The calculator also shows generated terms and step-by-step calculations so you can see how the result was obtained.
How to Use the Sequence Calculator
Choose the mode that matches the information you already have.
| Calculator Mode | Use It When |
|---|---|
| Analyze Sequence | You have several terms and want to identify a common pattern |
| Arithmetic | Each term changes by the same difference |
| Geometric | Each term is multiplied by the same ratio |
| Explicit Formula | You want a direct formula for finding any term |
| Recursive | Each new term depends on one or more previous terms |
To calculate a sequence:
- Select the appropriate calculator mode.
- Enter the sequence terms or known values.
- Enter the term number (n) if you want to find a specific term.
- Click the calculate or analyze button.
- Review the detected pattern, formula, requested term, sum, generated terms, and calculation steps.
If you are not sure whether a sequence is arithmetic or geometric, start with Analyze Sequence.
What Is a Number Sequence?
A number sequence is an ordered list of values called terms. Each term has a specific position in the sequence.
For example:
[
3,\ 7,\ 11,\ 15,\ 19,\ldots
]
Here:
- (a_1=3)
- (a_2=7)
- (a_3=11)
- (a_n) represents the term in position (n)
A sequence can follow a simple rule, such as adding the same number each time, multiplying by the same value, or using previous terms to generate the next one.
The rule matters because it determines which formula should be used to calculate later terms.
Arithmetic Sequences
An arithmetic sequence has a constant common difference between consecutive terms.
Consider:
[
4,\ 9,\ 14,\ 19,\ 24,\ldots
]
Subtract one term from the next:
[
9-4=5
]
[
14-9=5
]
[
19-14=5
]
The common difference is therefore:
[
d=5
]
Arithmetic Sequence Formula
The nth term of an arithmetic sequence can be calculated with:
[
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 of the term
Arithmetic Sequence Example
Find the 10th term of:
[
4,\ 9,\ 14,\ 19,\ldots
]
The first term is:
[
a_1=4
]
The common difference is:
[
d=5
]
Use the formula:
[
a_n=a_1+(n-1)d
]
Substitute (n=10):
[
a_{10}=4+(10-1)(5)
]
[
a_{10}=4+45
]
[
a_{10}=49
]
So the 10th term is 49.
Sum of an Arithmetic Sequence
Sometimes you need the total of the first (n) terms rather than one individual term.
One useful arithmetic series formula is:
[
S_n=\frac{n(a_1+a_n)}{2}
]
For the first 10 terms of the previous example:
[
S_{10}=\frac{10(4+49)}{2}
]
[
S_{10}=265
]
The Arithmetic mode calculates the nth term and the sum together.
Geometric Sequences
A geometric sequence has a constant common ratio between consecutive terms.
For example:
[
5,\ 10,\ 20,\ 40,\ 80,\ldots
]
Divide each term by the previous term:
[
\frac{10}{5}=2
]
[
\frac{20}{10}=2
]
[
\frac{40}{20}=2
]
The common ratio is:
[
r=2
]
Geometric Sequence Formula
The nth term of a geometric sequence is:
[
a_n=a_1r^{n-1}
]
where:
- (a_n) is the nth term
- (a_1) is the first term
- (r) is the common ratio
- (n) is the term position
Geometric Sequence Example
Find the 8th term of:
[
5,\ 10,\ 20,\ 40,\ldots
]
Here:
[
a_1=5
]
and:
[
r=2
]
Use the geometric formula:
[
a_8=5(2)^{8-1}
]
[
a_8=5(2)^7
]
[
a_8=640
]
So the 8th term is 640.
Sum of a Geometric Sequence
For a finite geometric sequence where (r\neq1), the sum of the first (n) terms can be calculated with:
[
S_n=\frac{a_1(1-r^n)}{1-r}
]
The calculator applies this formula automatically after you enter the first term, common ratio, and number of terms.
Infinite Geometric Sum
Some geometric sequences can also produce a finite infinite-series sum.
When:
[
|r|<1
]
the infinite sum is:
[
S_\infty=\frac{a_1}{1-r}
]
For example:
[
8,\ 4,\ 2,\ 1,\ldots
]
has a common ratio of:
[
r=\frac{1}{2}
]
Because (|r|<1), the associated infinite geometric series has a finite sum.
If the absolute value of the common ratio is 1 or greater, the calculator does not report a finite infinite sum.
How to Find the Nth Term of a Sequence
The nth term is the value at a specific position in a sequence.
There is no single nth-term formula that works for every sequence. The correct method depends on the pattern.
| Sequence Type | Nth-Term Formula |
|---|---|
| Arithmetic | (a_n=a_1+(n-1)d) |
| Geometric | (a_n=a_1r^{n-1}) |
For an arithmetic sequence, identify the first term and common difference.
For a geometric sequence, identify the first term and common ratio.
If you only know the existing terms, enter them into the Analyze Sequence mode first. The calculator checks whether the terms match one of its supported patterns and then calculates the requested term when possible.
Explicit Formula vs Recursive Formula
Explicit and recursive formulas describe sequences in different ways.
An explicit formula calculates a term directly from its position.
A recursive formula generates a term using one or more earlier terms.
Consider the arithmetic sequence:
[
3,\ 7,\ 11,\ 15,\ldots
]
Its common difference is 4.
Explicit Formula
An explicit rule can be written as:
[
a_n=3+4(n-1)
]
If you want the 50th term, you can substitute (n=50) directly without calculating terms 2 through 49 first.
Recursive Formula
The same sequence can be written recursively as:
[
a_1=3
]
[
a_n=a_{n-1}+4
]
This means the sequence starts at 3, and each new term is found by adding 4 to the previous term.
| Formula Type | How It Works | Example |
|---|---|---|
| Explicit | Uses the position (n) directly | (a_n=3+4(n-1)) |
| Recursive | Uses previous term values | (a_n=a_{n-1}+4) |
The calculator displays both forms when they are available.
How the Calculator Detects a Sequence Pattern
When you enter a list of existing terms, the calculator checks several common mathematical relationships.
It does not simply guess the next number from the visual appearance of the sequence.
Constant First Differences
Consider:
[
2,\ 6,\ 10,\ 14,\ 18
]
The differences are:
[
4,\ 4,\ 4,\ 4
]
Because the first differences are constant, the calculator identifies this as an arithmetic sequence with:
[
d=4
]
Constant Ratios
Consider:
[
3,\ 9,\ 27,\ 81
]
The consecutive ratios are:
[
3,\ 3,\ 3
]
A constant ratio indicates a geometric sequence.
In this example:
[
r=3
]
Constant Second Differences
Some sequences are not arithmetic because their first differences are not constant.
Consider:
[
1,\ 4,\ 9,\ 16,\ 25
]
The first differences are:
[
3,\ 5,\ 7,\ 9
]
The second differences are:
[
2,\ 2,\ 2
]
A constant second difference can indicate a quadratic pattern.
For this example, the familiar rule is:
[
a_n=n^2
]
The Analyze Sequence mode can identify supported constant-second-difference patterns and calculate additional terms.
Fibonacci-Like Recurrence
A sequence may also depend on previous terms rather than a constant difference or ratio.
For example:
[
2,\ 3,\ 5,\ 8,\ 13,\ 21
]
Each term after the first two is the sum of the previous two:
[
2+3=5
]
[
3+5=8
]
[
5+8=13
]
The recursive rule is:
[
a_n=a_{n-1}+a_{n-2}
]
The calculator can detect this common Fibonacci-like relationship.
Can a Number Sequence Have More Than One Pattern?
Yes. A short finite sequence does not always determine one unique mathematical rule.
For example, a small set of terms may fit a simple arithmetic or polynomial pattern, but more complicated formulas could also be constructed to reproduce those same initial values.
That is why the calculator reports detected or supported patterns rather than claiming that every result is the only possible mathematical rule.
For typical arithmetic, geometric, quadratic, and Fibonacci-like sequences, the detected pattern is often the intended one. For unusual puzzle sequences or patterns based on external ideas, the intended rule may require additional context.
Sequence vs Series
A sequence and a series are related, but they are not the same thing.
A sequence is an ordered list of terms:
[
2,\ 4,\ 6,\ 8
]
A series is the sum of those terms:
[
2+4+6+8
]
The result of that series is:
[
20
]
The calculator works primarily with sequences, but the Arithmetic and Geometric modes also calculate relevant finite sums.
Sequence Calculator Examples
Here are several common patterns the calculator can analyze.
| Sequence | Pattern | Rule | Next Term |
|---|---|---|---|
| 3, 7, 11, 15 | Arithmetic | Add 4 | 19 |
| 2, 6, 18, 54 | Geometric | Multiply by 3 | 162 |
| 1, 4, 9, 16 | Quadratic | (n^2) | 25 |
| 1, 1, 2, 3, 5 | Fibonacci-like | Add previous two terms | 8 |
Example 1: Find the 12th Term of an Arithmetic Sequence
Consider:
[
7,\ 12,\ 17,\ 22,\ldots
]
The common difference is:
[
d=5
]
Use:
[
a_n=a_1+(n-1)d
]
Substitute the values:
[
a_{12}=7+(12-1)(5)
]
[
a_{12}=7+55
]
[
a_{12}=62
]
The 12th term is 62.
Example 2: Find the 7th Term of a Geometric Sequence
Consider:
[
4,\ 12,\ 36,\ 108,\ldots
]
The common ratio is:
[
r=3
]
Use:
[
a_n=a_1r^{n-1}
]
Then:
[
a_7=4(3)^6
]
[
a_7=2916
]
The 7th term is 2916.
Frequently Asked Questions
What does a sequence calculator do?
A sequence calculator helps analyze number patterns and calculate terms from known sequence information. Depending on the mode, it can identify common arithmetic, geometric, quadratic, or Fibonacci-like patterns, calculate the nth term, generate additional terms, find sums, and display explicit or recursive formulas.
How do I find the nth term of a sequence?
First identify the sequence type. For an arithmetic sequence, use (a_n=a_1+(n-1)d). For a geometric sequence, use (a_n=a_1r^{n-1}). You can also enter the known values into the calculator and select the required term number.
How do I know if a sequence is arithmetic or geometric?
Check consecutive terms. If the difference between consecutive terms stays constant, the sequence is arithmetic. If the ratio between consecutive nonzero terms stays constant, the sequence is geometric.
What is an explicit formula for a sequence?
An explicit formula calculates a term directly from its position (n). For example, (a_n=5+3(n-1)) gives any term in an arithmetic sequence without requiring you to calculate all previous terms.
What is a recursive formula?
A recursive formula defines a term using one or more earlier terms. For example, (a_n=a_{n-1}+4) means each new term is four greater than the previous term. Recursive formulas normally also require one or more starting values.
How do I calculate the sum of an arithmetic sequence?
If you know the first term, last term, and number of terms, you can use:
[
S_n=\frac{n(a_1+a_n)}{2}
]
The Arithmetic mode calculates the sum automatically from the values you enter.
How do I calculate the sum of a geometric sequence?
For a finite geometric sequence with (r\neq1), use:
[
S_n=\frac{a_1(1-r^n)}{1-r}
]
Enter the first term, common ratio, and number of terms in the Geometric mode to calculate the result.
Can the same number sequence have more than one possible pattern?
Yes. A finite list of terms can sometimes be described by more than one mathematical rule. The calculator checks for common supported relationships, such as constant differences, constant ratios, constant second differences, and Fibonacci-like recurrences. It does not claim that a detected rule is always the only possible interpretation.