Number Sequence Calculator – Arithmetic, Geometric, and Fibonacci Sequences

  • Adam
  • November 4, 2025

Free online number sequence calculator for arithmetic sequences, geometric sequences, and Fibonacci sequences. Find nth term, calculate sums, and generate sequences with step-by-step solutions.

Number Sequence Calculator

Advanced Tool for Arithmetic, Geometric, and Fibonacci Sequences

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Arithmetic Sequence Calculator

Definition: aₙ = a₁ + f × (n-1)

Geometric Sequence Calculator

Definition: aₙ = a × rⁿ⁻¹

Fibonacci Sequence Calculator

Definition: a₀=0; a₁=1; aₙ = aₙ₋₁ + aₙ₋₂

What are Number Sequences?

A number sequence is an ordered list of numbers following a specific pattern or rule. Each number in the sequence is called a term. Sequences appear throughout mathematics, physics, finance, and nature. Understanding sequences helps solve problems involving patterns, predictions, and growth.

There are three main types: Arithmetic sequences have constant difference between consecutive terms. Geometric sequences have constant ratio between consecutive terms. Fibonacci sequences follow the pattern where each term is the sum of the previous two terms. Each type has unique properties and applications.

Sequences are fundamental to mathematics and science. They model population growth, investment returns, radioactive decay, architectural patterns, and countless natural phenomena. This calculator helps you find specific terms, calculate sums, and understand the mathematics behind these essential patterns.

Key Concept: Every sequence follows a specific rule or formula. Once you identify the pattern (difference or ratio), you can predict any term without calculating all previous ones.

Key Features & Capabilities

This comprehensive sequence calculator provides complete analysis for all three sequence types:

➕ Arithmetic Sequences Find any term with constant difference
✕ Geometric Sequences Find any term with constant ratio
🔢 Fibonacci Sequences Generate Fibonacci terms and series
🎯 Find nth Term Calculate any position directly
📊 Sum Calculation Calculate sum of sequences
📈 Generate Sequence Display all terms up to n
📋 Step-by-Step Detailed calculation breakdown
📊 Multiple Statistics Mean, range, and analysis
📋 Copy Function One-click copy to clipboard
🎓 Educational Learn sequence mathematics
⚡ Real-Time Instant calculations
📱 Responsive Works on all devices

How to Use This Calculator

Step-by-Step Guide

  1. Choose Sequence Type: Select Arithmetic, Geometric, or Fibonacci based on your sequence type.
  2. Enter Initial Values: For arithmetic/geometric, enter first term and common difference/ratio. For Fibonacci, just enter the position.
  3. Specify Position: Enter which term (n) you want to find. For Fibonacci, this is the only input needed.
  4. Click Calculate: Press Calculate to compute the nth term and generate the sequence.
  5. Review Results: See the nth term value, complete sequence, and sum of all terms.
  6. Study Calculation: Understand step-by-step how the formula was applied.
  7. Analyze Statistics: View mean, range, and other sequence properties.
  8. Copy or Clear: Use Copy for results or Clear to calculate a new sequence.

Tips for Accurate Use

  • Identify Pattern: Arithmetic has constant difference; geometric has constant ratio. Make sure you identify the correct type.
  • Use Decimals: The calculator supports decimal values for all sequence types.
  • Negative Values: Common difference/ratio can be negative for decreasing sequences.
  • Large n Values: Geometric sequences grow very quickly; large n may produce very large numbers.
  • Fibonacci Limit: Calculator supports up to position 100 for Fibonacci.

Complete Formulas Guide

Arithmetic Sequence

Finding the nth Term
aₙ = a₁ + (n - 1) × d

Where:
aₙ = nth term
a₁ = first term
d = common difference
n = position

Example: First term = 2, difference = 5, find 20th term
a₂₀ = 2 + (20-1) × 5 = 2 + 95 = 97

Arithmetic Sum

Sum of First n Terms
S = n/2 × (a₁ + aₙ)
or S = n/2 × (2a₁ + (n-1)d)

Sum of all terms up to position n
Example above: S₂₀ = 20/2 × (2 + 97) = 10 × 99 = 990

Geometric Sequence

Finding the nth Term
aₙ = a × r^(n-1)

Where:
aₙ = nth term
a = first term
r = common ratio
n = position

Example: First term = 2, ratio = 5, find 12th term
a₁₂ = 2 × 5^(12-1) = 2 × 5^11 = 2 × 48,828,125 = 97,656,250

Geometric Sum

Sum of First n Terms
S = a × (1 - r^n) / (1 - r) (when r ≠ 1)

Sum of all terms up to position n
For r > 1, terms grow exponentially
Note: When |r| < 1, sum approaches a/(1-r)

Fibonacci Sequence

Fibonacci Definition
F₀ = 0
F₁ = 1
Fₙ = Fₙ₋₁ + Fₙ₋₂ (for n ≥ 2)

Each term is sum of previous two
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...

Sequence Types Explained

Arithmetic Sequences

Arithmetic sequences have constant difference between consecutive terms. Example: 2, 5, 8, 11, 14 (difference = 3). The difference can be positive (increasing), negative (decreasing), or zero (constant). They grow linearly—each step adds the same amount. Used in modeling uniform growth like simple interest.

Geometric Sequences

Geometric sequences have constant ratio between consecutive terms. Example: 2, 6, 18, 54, 162 (ratio = 3). The ratio determines growth rate. Ratios greater than 1 show exponential growth; between 0 and 1 show decay. Geometric sequences model phenomena like compound interest and radioactive decay.

Fibonacci Sequences

Fibonacci sequences where each term equals the sum of previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... Appears throughout nature (flower petals, spiral shells, tree branches). The ratio between consecutive terms approaches the golden ratio (≈1.618). Named after Leonardo Fibonacci who introduced it in 1202.

Applications

Arithmetic: Rent increases, age at each year, linear depreciation. Geometric: Compound interest, population growth, half-life decay. Fibonacci: Nature patterns, stock analysis, algorithm design, data compression.

Worked Examples

Example 1: Arithmetic Sequence

Problem: First term = 2, common difference = 5. Find 20th term and sum of first 20 terms.

Solution:
Find 20th term: a₂₀ = 2 + (20-1) × 5 = 2 + 95 = 97

Sum of 20 terms: S₂₀ = 20/2 × (2 + 97) = 10 × 99 = 990

Sequence: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87, 92, 97

Example 2: Geometric Sequence

Problem: First term = 2, common ratio = 3. Find 8th term.

Solution:
a₈ = 2 × 3^(8-1) = 2 × 3^7 = 2 × 2,187 = 4,374

Sequence: 2, 6, 18, 54, 162, 486, 1,458, 4,374

Note: Grows rapidly compared to arithmetic

Example 3: Fibonacci Sequence

Problem: Find the 10th Fibonacci number.

Solution:
F₀ = 0
F₁ = 1
F₂ = 0 + 1 = 1
F₃ = 1 + 1 = 2
F₄ = 1 + 2 = 3
F₅ = 2 + 3 = 5
F₆ = 3 + 5 = 8
F₇ = 5 + 8 = 13
F₈ = 8 + 13 = 21
F₉ = 13 + 21 = 34
F₁₀ = 21 + 34 = 55

Example 4: Arithmetic Sum Application

Problem: Your salary starts at $30,000 and increases $2,000 per year. Total earnings after 10 years?

Solution:
Year 1: $30,000
Year 10: 30,000 + (10-1) × 2,000 = $48,000

Total earnings (arithmetic sum):
S = 10/2 × (30,000 + 48,000)
S = 5 × 78,000 = $390,000

Example 5: Geometric Growth Application

Problem: Investment doubles every year. Starting with $1,000, what's it worth after 10 years?

Solution:
a₁ = $1,000 (initial investment)
r = 2 (doubles)
a₁₀ = 1,000 × 2^(10-1) = 1,000 × 512 = $512,000

Shows power of exponential growth

Frequently Asked Questions

❓ How do I identify which sequence type I have?
Subtract consecutive terms. If difference is constant → arithmetic. If ratio is constant → geometric. If sum of previous two → Fibonacci. Check first 3-4 terms to confirm.
❓ Can arithmetic sequences decrease?
Yes. If common difference is negative, sequence decreases. Example: 10, 7, 4, 1, -2... has d = -3. Still arithmetic.
❓ Can geometric sequences have negative ratio?
Yes. Negative ratio causes alternating signs. Example: 2, -6, 18, -54... has r = -3. Terms oscillate between positive and negative.
❓ What happens with geometric ratio between 0 and 1?
Sequence decreases toward zero. Example: 100, 50, 25, 12.5... has r = 0.5. Series converges to finite sum: S = a/(1-r).
❓ Where is Fibonacci found in nature?
Fibonacci appears in flower petals, spiral shells, pine cones, tree branches, galaxy spirals. Also in financial markets, stock prices, and population dynamics.
❓ What's the golden ratio relation to Fibonacci?
Golden ratio (φ ≈ 1.618) is limit of ratios of consecutive Fibonacci numbers. As n increases, Fₙ₊₁/Fₙ → φ. Appears in art, architecture, and nature.
❓ Can I have both arithmetic and geometric properties?
Rarely. A sequence is either arithmetic, geometric, or neither. The only sequence both are is the constant sequence (d = 0, r = 1). Example: 5, 5, 5, 5...
❓ How do I find a sequence formula from just data?
Check differences (arithmetic) or ratios (geometric). For arithmetic: d = (a₂ - a₁), then a₁. For geometric: r = a₂/a₁, then a₁.
❓ What's the difference between sequence and pattern?
Sequence is specific mathematical ordering with explicit formulas. Pattern is general regularity. All sequences follow patterns, but not all patterns are sequences.
❓ When would I use these in real life?
Arithmetic: rent raises, age progression. Geometric: compound interest, investment growth, decay rates. Fibonacci: population modeling, algorithm analysis, nature prediction.

Start Calculating Sequences

Whether you're studying mathematics, analyzing patterns, modeling growth, or exploring natural phenomena, this comprehensive number sequence calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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