How to Use the Number Sequence Calculator
This suite provides four dedicated calculation modes designed for pattern detection, progression analysis, recurrence modeling, and custom function evaluation:
In the Sequence Pattern & Formula Finder, paste or type your known terms separated by commas, spaces, tabs, or line breaks (e.g. 2, 5, 10, 17, 26). Enter your target index n to calculate distant term values and partial sums.
Select Arithmetic or Geometric, enter the first term a₁, input the common difference d or ratio r, and specify term count n to compute the explicit term and closed-form sum.
Choose standard Fibonacci, Lucas, or custom initial seeds (a₁, a₂). Specify your desired index n to compute the recurrence value, Binet-formula approximation, and cumulative sum.
Enter any explicit algebraic function of n (such as n^2 + 1, 2^n - 1, or 1/n). Specify the target term and generation range to evaluate exact term values and analyze supported series convergence.
What Is a Number Sequence?
A number sequence is an ordered list of numbers, called terms, arranged according to a mathematical rule. The position of each term is indexed by a positive integer n (starting at n = 1), and the value at position n is written as aₙ.
For example, 3, 7, 11, 15, 19, ... is an ordered progression where every term adds 4 to the previous term. In contrast, 2, 6, 18, 54, 162, ... multiplies each preceding term by 3.
How to Find the Next Term in a Number Sequence
To find the next term in an unfamiliar sequence, test the consecutive terms systematically:
- Check consecutive differences: Subtract each term from the next (ak+1 − ak). If the difference is constant, the sequence is arithmetic.
- Check consecutive ratios: Divide each term by the previous one (ak+1 / ak). If the ratio is constant, the sequence is geometric.
- Examine finite differences: If first differences change, take second (Δ²) or third (Δ³) differences to check for polynomial (quadratic or cubic) patterns.
- Test additive recurrence: Check if terms equal the sum of previous terms (an = an−1 + an−2), such as in Fibonacci-style sequences.
- Inspect alternating operations: Look for repeating interleaved steps, such as alternating addition and multiplication.
nth Term Calculator
What Does the nth Term Mean?
In mathematics, the nth term (commonly denoted as an, un, or f(n)) represents the general rule and the specific numerical value occupying position n in an ordered sequence.
Without an explicit nth-term formula, finding the value of a distant term (such as the 50th or 100th term) requires manually listing and computing every preceding term step-by-step. An nth-term formula provides an immediate algebraic shortcut: by substituting the desired position index n directly into the equation, you can calculate the exact value of any term instantly.
Arithmetic nth Term Formula
For any arithmetic sequence with initial term a₁ and constant common difference d, each consecutive step adds d. Reaching term n requires taking (n − 1) additive steps:
Geometric nth Term Formula
For a geometric sequence with initial term a₁ and constant common ratio r, each term multiplies the predecessor by r. Finding the nth term multiplies a₁ by r exactly (n − 1) times:
Polynomial nth Term Formula
When differences between terms are not constant, sequences often follow polynomial laws where the position index n is raised to powers:
- Quadratic Sequences: When second differences (Δ²) are constant, the closed form is quadratic: an = An² + Bn + C.
- Cubic Sequences: When third differences (Δ³) are constant, the closed form is cubic: an = An³ + Bn² + Cn + D.
Custom Function / Sequence Formula
Many sequences are defined by explicit algebraic functions f(n), treating n as an independent variable. You can enter any custom explicit expression in terms of n into the calculator to compute both individual terms and cumulative series sums on demand.
Worked nth-Term Examples
When a sequence formula is already known, finding a specific term only requires substituting the target value of n and simplifying the expression.
Substitute n = 7:
Substitute n = 6:
Arithmetic Sequence Calculator
What Is an Arithmetic Sequence?
An arithmetic sequence (or arithmetic progression, AP) is a sequence in which each term after the first is formed by adding a constant value, known as the common difference (d), to the preceding term. Because the same amount is added at every step, arithmetic sequences exhibit linear discrete growth.
Common Difference
The common difference d is the difference between any term and its immediate predecessor:
For example, in 3, 7, 11, 15, 19, ..., d = 7 − 3 = 4. When a sequence decreases, such as 20, 17, 14, 11, ..., d = 17 − 20 = −3.
nth Term of an Arithmetic Sequence
To find the nth term without listing every preceding value, use:
For 3, 7, 11, 15, ... with a₁ = 3, d = 4, the 10th term is a₁₀ = 3 + 9(4) = 39.
Sum of an Arithmetic Sequence
The partial sum of the first n terms (Sₙ) is the average of the first and last term multiplied by the number of terms:
For example, the sum of the first 10 terms of 3, 7, 11, 15, ... is S₁₀ = 5(3 + 39) = 210.
Geometric Sequence Calculator
What Is a Geometric Sequence?
A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant common ratio r.
- When |r| > 1, terms grow rapidly in magnitude.
- When |r| < 1, terms diminish toward zero.
- When r < 0, consecutive terms alternate in sign (+, −, +, −).
Common Ratio
The common ratio r is found by dividing any term by its preceding term:
For example, in 2, 6, 18, 54, ..., r = 6 / 2 = 3. In 2, −4, 8, −16, ..., r = (−4) / 2 = −2.
nth Term Formula
The general formula for the nth term of a geometric sequence is:
For 2, 6, 18, 54, 162, ... (a₁ = 2, r = 3), the 6th term is a₆ = 2 · 3⁵ = 486.
Sum of a Geometric Sequence
The partial sum of the first n terms (Sₙ) of a geometric sequence is:
Infinite Geometric Series
An infinite geometric series converges to a finite sum if and only if |r| < 1:
When |r| ≥ 1, the infinite series diverges without settling on a finite limit.
Finite Differences for Quadratic and Cubic Sequences
Finite differences provide a systematic algebraic test for sequences whose consecutive differences are not constant. Taking repeated differences reveals the underlying polynomial degree:
- Constant First Differences (Δ¹): Linear / arithmetic sequence (an = An + B).
- Constant Second Differences (Δ²): Quadratic sequence (an = An² + Bn + C), where 2A = Δ².
- Constant Third Differences (Δ³): Cubic sequence (an = An³ + Bn² + Cn + D), where 6A = Δ³.
First differences (Δ¹): 3, 5, 7, 9
Second differences (Δ²): 2, 2, 2 (constant!)
Since 2A = 2, we find A = 1. Fitting the remaining coefficients gives the governing rule an = n² + 1.
Fibonacci and Recurrence Sequence Calculator
A recurrence relation defines each term using one or more preceding terms. In a second-order additive recurrence, every term after the initial seeds equals the sum of the two preceding values:
Fibonacci Sequence
The standard Fibonacci sequence begins with seed values F₁ = 1, F₂ = 1 (or F₀ = 0, F₁ = 1), generating 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .... Consecutive ratios converge to the golden ratio φ ≈ 1.6180339.
Lucas Sequence
The Lucas sequence uses the same additive relation (Ln = Ln−1 + Ln−2) but starts with seeds L₁ = 1, L₂ = 3 (or L₀ = 2, L₁ = 1), producing 2, 1, 3, 4, 7, 11, 18, 29, 47, ....
Custom Fibonacci-Type Recurrence
By selecting any two custom starting seeds (such as 2, 3 yielding 2, 3, 5, 8, 13, 21, ...), the recurrence engine evaluates generalized second-order additive sequences directly.
Sequence vs. Series
A sequence is an ordered list of individual numbers (a₁, a₂, a₃, ...), whereas a series is the cumulative sum of those numbers (a₁ + a₂ + a₃ + ...).
Partial Sum of a Sequence
The partial sum Sₙ represents the sum of the first n terms:
Sum of an Arithmetic Sequence
Calculated using Sn = (n / 2)(a₁ + an).
Sum of a Geometric Sequence
Calculated using Sn = a₁(1 − rn) / (1 − r) when r ≠ 1.
Infinite Geometric Series
Convergence analysis applies the supported mathematical tests for the supplied series. For an infinite geometric series, convergence occurs when |r| < 1. For other supported functions, the calculator may identify divergence using the nth-term test when the terms do not approach zero.
Sequence Formula Calculator
Evaluate a Known Formula
When the governing algebraic equation is already known, the custom-function mode evaluates the formula directly without requiring pattern inference. You can input expressions in terms of n, such as polynomials, powers, rational functions, or alternating signs, to calculate any specific term and inspect series sums instantly.
Find a Formula From a Sequence
When starting from observed numbers rather than a known equation, the calculator inspects the terms to infer a governing rule. For supported patterns, the calculator identifies the governing pattern and derives the corresponding formula.
Arithmetic Pattern
Constant first differences lead to an = a₁ + (n − 1)d.
Geometric Pattern
Constant ratios between terms lead to an = a₁ · rn−1.
Polynomial Pattern
Constant higher-order finite differences yield quadratic (An² + Bn + C) or cubic (An³ + Bn² + Cn + D) equations.
Fibonacci-Type Pattern
Terms formed by adding the previous two values yield additive recurrence relations.
Can a Number Sequence Have More Than One Answer?
A finite list of numbers does not always uniquely determine a single mathematical rule. For example, the terms 5, 8 could continue as 11 under an arithmetic rule (d = 3), but infinitely many other polynomial, exponential, or periodic formulas can also generate those initial terms.
The calculator therefore reports supported and detected mathematical patterns based on standard algebraic models, rather than claiming that every continuation is mathematically unique.
How the Number Sequence Calculator Finds a Pattern
The calculator analyzes the supplied terms using supported mathematical tests rather than treating sequences as a black box:
Worked Number Sequence Examples
Arithmetic Sequence
Consider: 4, 9, 14, 19, 24, ...
Common difference: d = 5. Formula: an = 4 + 5(n − 1) = 5n − 1.
Geometric Sequence
Consider: 3, 12, 48, 192, ...
Common ratio: r = 4. Formula: an = 3 · 4n−1.
nth Term From a Formula
Given f(n) = 3 − 5(n − 1), find the 7th term:
Quadratic Sequence
Consider: 3, 8, 15, 24, 35, ...
First diffs: 5, 7, 9, 11. Second diffs: 2, 2, 2 (constant).
Cubic Sequence
Consider: 1, 8, 27, 64, 125, ...
Terms are perfect cubes 1³, 2³, 3³, 4³, 5³.
Fibonacci-Type Recurrence
Consider: 2, 3, 5, 8, 13, 21, ...
Each term equals the sum of the preceding two values:
Alternating Pattern
Consider: 2 → 5 → 10 → 13 → 26 → ? with alternating operations (+3, ×2, +3, ×2, ...).
This example illustrates how to solve an alternating pattern manually; alternating-operation rules are not necessarily part of the calculator's standard automatic pattern-detection models.
Infinite Geometric Series
Consider: 81, 27, 9, 3, 1, ... (a₁ = 81, r = 1/3).
Limitations of Number Sequence Pattern Detection
Not every number sequence follows an arithmetic, geometric, polynomial, or standard recurrence model. Some sequences involve:
- alternating operations (+3, ×2);
- piecewise or conditional definitions;
- prime numbers and prime gaps;
- higher-order recursive rules involving three or more previous terms;
- periodic or modular arithmetic cycles;
- factorials, partitions, or digit-manipulation rules.
An "unclassified" result indicates that the supplied terms do not match the supported linear, geometric, polynomial, or second-order recurrence families, rather than implying that no mathematical rule exists. Providing additional terms often clarifies the structure.
Where Number Sequences Are Used
Number sequences appear across mathematics, science, and computing. Arithmetic progressions model fixed intervals, depreciation schedules, and uniform sampling rates. Geometric series calculate compound interest, depreciation, population growth, and loan amortizations. Polynomial and difference patterns describe physical trajectories, accelerated rates, and geometry formulas. Recurrence relations power dynamic programming, recursive algorithms, and branching simulations.
Frequently Asked Questions
A number sequence calculator analyzes a list of numbers to identify supported mathematical patterns, calculate terms, and evaluate sequence formulas. CalcPlatform can analyze arithmetic, geometric, quadratic, cubic, and Fibonacci-type patterns, as well as evaluate custom sequence functions and calculate supported sums.
Enter the known terms into the sequence calculator and examine the detected pattern. Depending on the sequence, the next term may be determined from a constant difference, constant ratio, finite differences, or a recurrence pattern. A finite list can sometimes have more than one mathematically valid continuation, so the detected pattern should be treated as the rule being applied rather than the only possible answer.
Enter the known sequence terms into the pattern or formula calculator. For supported patterns, the calculator can identify a mathematical rule such as an arithmetic, geometric, or polynomial relationship. You can also enter a formula directly in the custom-function mode to generate or evaluate sequence terms.
The nth term is the term at position n in a sequence. For example, the 1st term is a₁, the 2nd term is a₂, and the 7th term is a₇. The nth-term formula depends on the type of sequence. For an arithmetic sequence, for example:
aₙ = a₁ + (n − 1)d
where a₁ is the first term and d is the common difference.
For an arithmetic sequence, calculate the common difference between consecutive terms and use:
aₙ = a₁ + (n − 1)d
For example, for 4, 7, 10, 13, ..., the common difference is 3. The 10th term is:
a₁₀ = 4 + (10 − 1)(3) = 31
For a geometric sequence, first find the common ratio between consecutive terms. Then use:
aₙ = a₁rⁿ⁻¹
where a₁ is the first term and r is the common ratio.
For example, for 3, 6, 12, 24, ..., the common ratio is 2, so the nth term is:
aₙ = 3 × 2ⁿ⁻¹
Yes. The calculator can use finite differences to analyze supported quadratic and cubic patterns. A constant second difference is associated with a quadratic sequence, while a constant third difference is associated with a cubic sequence.
The calculator supports Fibonacci-type and Lucas-style recurrence sequences. It can work with recurrence relationships and different starting values rather than being limited to only the standard Fibonacci sequence.
Yes. The calculator supports partial sums for supported sequence types. For arithmetic and geometric sequences, standard sum formulas can be used to calculate the total of the first n terms. It also supports infinite geometric-series analysis when the required convergence conditions are satisfied.
Yes. Use the custom-function mode to enter a mathematical expression involving n, such as:
n² + 1
2ⁿ − 1
3n + 5
1/n
The calculator can then evaluate the function for sequence terms and analyze supported series behavior.
A finite list of numbers does not always uniquely determine a mathematical rule. Different formulas can produce the same initial terms but give different later terms. For this reason, sequence pattern problems should be interpreted according to the simplest or intended rule supported by the problem and the calculator's detected pattern.
Yes. For supported sequence patterns, the calculator provides the relevant intermediate information, such as differences, ratios, formulas, generated terms, or recurrence calculations, so the result can be checked rather than treated as a black-box answer.