Number Sequence Calculator
Generate arithmetic, geometric, or Fibonacci-style number sequences — with every term, the running sum, and the nth-term formula.
Sum of First 12 Terms
366
Example
Starting from 3, this sequence's first 12 terms sum to 366.
nth-Term Formula
aₙ = 3 + (n − 1) × 5
What is a Number Sequence?
A number sequence is an ordered list of numbers that follows a consistent rule from one term to the next. This calculator covers three of the most common patterns: arithmetic (each term adds a fixed amount), geometric (each term multiplies by a fixed ratio), and Fibonacci-style (each term is the sum of the two before it).
Term value by position in the sequence
Full Term List
| Term # | Value |
|---|---|
| 1 | 3 |
| 2 | 8 |
| 3 | 13 |
| 4 | 18 |
| 5 | 23 |
| 6 | 28 |
| 7 | 33 |
| 8 | 38 |
| 9 | 43 |
| 10 | 48 |
| 11 | 53 |
| 12 | 58 |
How Do These Sequences Work?
Each sequence type defines the next term using only a simple rule and the term(s) before it — no pattern requires looking arbitrarily far back, which is what makes them predictable and easy to extend indefinitely.
Arithmetic Sequences
An arithmetic sequence adds the same fixed amount (the "common difference") to get from one term to the next — 3, 8, 13, 18 is arithmetic with a common difference of 5. Because the growth is constant rather than compounding, these sequences grow linearly: plotting the terms produces a straight line.
Geometric Sequences
A geometric sequence multiplies by the same fixed ratio each time — 2, 6, 18, 54 is geometric with a common ratio of 3. This produces exponential growth (or decay, if the ratio is between −1 and 1), which is why compound interest, population growth models, and radioactive decay are all real-world geometric sequences in disguise.
Fibonacci-Style Sequences
The classic Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13...) starts at 0 and 1 and adds the two most recent terms to get the next. This calculator generalizes that rule to any starting pair — changing the first two terms produces a "Fibonacci-style" or "Lucas-style" sequence that still follows the same additive rule, just from a different starting point.
Example — Your Current Inputs
Starting from 3, this sequence's first 12 terms sum to 366.
Additional Example — The Golden Ratio
Dividing each Fibonacci number by the one before it (5÷3, 8÷5, 13÷8...) converges toward approximately 1.618 — the golden ratio (φ). This convergence happens regardless of the sequence's starting values, as long as each term is the sum of the previous two, which is a striking example of a pattern that emerges from the *rule* of a sequence rather than from any specific starting numbers.
About These Parameters
- First Term (& Second Term)
- Where the sequence starts. Fibonacci-style sequences need two starting values since each new term depends on the previous two.
- Common Difference / Common Ratio
- The fixed amount added (arithmetic) or multiplied (geometric) at every step. A ratio between −1 and 1 (exclusive of 0) produces a sequence that shrinks toward zero, which is when a finite "sum to infinity" exists.
- Number of Terms
- How many terms to generate, capped at 30 to keep results readable and to avoid numeric overflow on fast-growing geometric sequences.
Frequently Asked Questions
What does "sum to infinity" mean for a geometric sequence?
When the common ratio's absolute value is less than 1, each new term is smaller than the last, and the running total converges toward a finite limit even though the sequence itself never ends. If the ratio's absolute value is 1 or greater, the sum grows without bound and no finite "sum to infinity" exists.
Can the common difference or ratio be negative?
Yes. A negative common difference produces a decreasing arithmetic sequence; a negative common ratio produces a geometric sequence that alternates between positive and negative terms while still growing (or shrinking) in magnitude.
Is every number sequence either arithmetic, geometric, or Fibonacci-style?
No — these are just three especially common patterns. Sequences can follow countless other rules (quadratic growth, alternating patterns, sequences defined by a formula with no simple term-to-term relationship, and more). These three are covered here because they show up most often in classrooms and real-world growth models.