Number Sequence Calculator

Complete Number Sequence Guide & Information

1. 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 or infinite. This calculator handles three of the most important types: arithmetic, geometric, and Fibonacci.

2. Arithmetic Sequence

An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. The difference between consecutive terms is always the same.

Nth term formula

where a₁ is the first term, d is the common difference, and n is the term number.

Sum of first n terms

or equivalently:

Properties

• Linear growth — terms increase by a constant amount
• The difference between any two consecutive terms is constant
• Graph of term values vs. term number forms a straight line
• Three consecutive terms a, b, c satisfy 2b = a + c

3. Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio.

Nth term formula

where a₁ is the first term, r is the common ratio, and n is the term number.

Sum of first n terms

Properties

• Exponential growth or decay
• The ratio of any two consecutive terms is constant
• If |r| > 1, terms grow without bound
• If |r| < 1, terms approach zero (infinite geometric series converges)
• Three consecutive terms a, b, c satisfy b² = a × c

4. Fibonacci Sequence

The Fibonacci sequence starts with 1, 1 (or sometimes 0, 1), and each subsequent term is the sum of the two preceding terms. It appears throughout nature, art, and mathematics.

Recurrence relation

Closed-form (Binet's formula)

Sum of first n terms

Properties

• Ratio of consecutive terms approaches the golden ratio φ ≈ 1.618
• Appears in nature: sunflower seeds, nautilus shells, tree branching
• Closely related to the golden rectangle and golden spiral
• Grows exponentially but slower than geometric sequences
• Cassini's identity: F_n+1 × F_n−1 − F_n² = (−1)ⁿ

5. Input & Control Definitions

• First number: The starting value a₁ of the sequence.
• Common difference (f): Constant value added to each term (arithmetic only).
• Common ratio (r): Constant multiplier for each term (geometric only).
• n^th number to obtain: How many terms to compute and which term to highlight.
• Calculate button: Computes the sequence, nth term, and sum.
• Clear button: Resets the input fields to empty.
• Sequence display: First several terms of the sequence.
• Nth value: The value of the term at position n.
• Sum: Sum of all terms from the first through the nth.
• Growth chart: Line chart visualizing how term values increase.
• Tabs: Switch between arithmetic, geometric and Fibonacci calculators.

6. Worked Examples

Example 1 — Arithmetic (default): a₁ = 2, d = 5, n = 20

• a₂₀ = 2 + 19 × 5 = 2 + 95 = 97
• Sum = 20 × (2 + 97) / 2 = 10 × 99 = 990

Example 2 — Geometric (default): a₁ = 2, r = 5, n = 12

• a₁₂ = 2 × 5¹¹ = 2 × 48,828,125 = 97,656,250
• Sum = 2 × (1 − 5¹²) / (1 − 5) = 122,070,312

Example 3 — Fibonacci (default): n = 10

• Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
• F₁₀ = 55
• Sum = F₁₂ − 1 = 144 − 1 = 143

7. Real-World Applications

• Finance: Simple interest (arithmetic), compound interest (geometric)
• Physics: Free fall distance, radioactive decay (geometric)
• Computer science: Algorithm analysis, Fibonacci heaps, binary search
• Biology: Population growth (geometric), phyllotaxis (Fibonacci)
• Economics: Annuities, geometric growth models
• Architecture & art: Golden ratio proportions, Fibonacci spirals
• Music: Rhythmic patterns and Fibonacci-based composition
• Everyday: Seating arrangements, stacking problems, pacing

8. Comparison Reference Table

Property	Arithmetic	Geometric	Fibonacci
Growth type	Linear	Exponential	Exponential (slower)
Constant	Difference d	Ratio r	Sum of previous two
Nth term formula	a₁ + (n−1)d	a₁ × rn−1	Recursive / Binet's
Sum formula	n(a₁+aₙ)/2	a₁(1−rⁿ)/(1−r)	Fn+2 − 1
Example n=10	2,7,12,...,47	2,10,50,...,3,906,250	1,1,2,...,55

9. Important Notes

• Arithmetic sequences always grow at a constant rate per step.
• Geometric sequences with r > 1 grow faster than any arithmetic sequence for large n.
• Fibonacci numbers grow roughly like φⁿ / √5 where φ ≈ 1.618.
• Very large n values may produce extremely large numbers.
• Negative common differences produce decreasing arithmetic sequences.
• Negative common ratios produce alternating positive/negative terms.
• For infinite geometric series with |r| < 1, total sum = a₁ / (1 − r).

10. Related Mathematical Concepts

• Series: Sum of the terms of a sequence
• Golden ratio (φ): Limit of Fibonacci term ratios
• Recurrence relation: Equation defining terms from previous terms
• Convergence: Whether an infinite series has a finite sum
• Binomial coefficients: Pascal's triangle contains Fibonacci numbers
• Continued fractions: Golden ratio has simplest continued fraction
• Linear recurrence: Generalization of arithmetic and Fibonacci sequences

11. References

1. Fibonacci, Leonardo. "Liber Abaci." 1202.
2. Euclid. "Elements." Book VI, Proposition 30 (golden ratio). ~300 BCE.
3. Knuth, Donald E. "Concrete Mathematics." Addison-Wesley. 1994.
4. Koshy, Thomas. "Fibonacci and Lucas Numbers with Applications." Wiley. 2001.
5. Stewart, Ian. "Mathematical Recreations: The Tales of a Not-So-Famous Number." Scientific American. 1995.
6. Dunlap, Richard A. "The Golden Ratio and Fibonacci Numbers." World Scientific. 1997.