Number Sequence Calculator

Generate arithmetic, geometric, and Fibonacci sequences, and see the sequence and its sum instantly.

Arithmetic Sequence

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Geometric Sequence

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Fibonacci Sequence

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Number Sequences: Complete Guide to Arithmetic, Geometric, and Fibonacci Series

Number sequences follow specific patterns, and understanding these patterns helps with algebra, series, growth models, and problem-solving.

This guide mirrors your other calculator pages with multiple boxes, rules, worked examples, applications, pitfalls, and practice problems, tailored to each sequence type supported by the calculator above.

What Is a Number Sequence?

A number sequence is an ordered list of numbers that follow a rule, such as adding a fixed difference, multiplying by a constant ratio, or adding the previous terms.

Key Terms
  • Term: Each individual number in the sequence (often written a₁, a₂, a₃, …).
  • Index n: Position of a term in the sequence, starting from 1.
  • Rule / formula: Description of how to get each term (explicit or recursive).
Example 1: Increasing by 3
2, 5, 8, 11, 14, … is an arithmetic sequence with common difference 3.
Example 2: Doubling each time
1, 2, 4, 8, 16, … is a geometric sequence with common ratio 2.
Sequence Type Pattern Next Terms
3, 6, 9, 12 Arithmetic +3 each step 15, 18
5, 10, 20, 40 Geometric ×2 each step 80, 160
0, 1, 1, 2, 3 Fibonacci Sum of previous two 5, 8
10, 7, 4, 1 Arithmetic −3 each step −2, −5
9, 3, 1, 1/3 Geometric ÷3 each step 1/9, 1/27

Arithmetic Sequences

In an arithmetic sequence, the difference between consecutive terms is constant, making it ideal for modeling linear growth or decay.

General formulas

n‑th term: aₙ = a₁ + (n−1)d
Sum of n terms: Sₙ = n/2 · [2a₁ + (n−1)d]

Example 1: Generate terms
a₁ = 2, d = 3, n = 5.
a₂ = 5, a₃ = 8, a₄ = 11, a₅ = 14 → sequence 2, 5, 8, 11, 14.
Example 2: Find a term and sum
a₁ = 4, d = 2, n = 10.
a₁₀ = 4 + 9·2 = 22.
S₁₀ = 10/2 · (4 + 22) = 5 · 26 = 130.
Parameter Meaning Example
a₁ First term 2 in 2, 5, 8, …
d Common difference 3 in 2, 5, 8, …
aₙ n‑th term a₄ = 11 for n = 4
Sₙ Sum of first n terms S₅ = 2+5+8+11+14

Geometric Sequences

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, which naturally models exponential growth or decay.

General formulas

n‑th term: aₙ = a₁ · rn−1
Sum (r ≠ 1): Sₙ = a₁ · (1 − rn) / (1 − r)

Example 1: Doubling sequence
a₁ = 1, r = 2, n = 6.
Terms: 1, 2, 4, 8, 16, 32.
Sum S₆ = 1 · (1 − 2⁶)/(1 − 2) = (1 − 64)/(−1) = 63.
Example 2: Fractional ratio
a₁ = 81, r = 1/3, n = 4.
Terms: 81, 27, 9, 3.
S₄ = 81 · (1 − (1/3)⁴)/(1 − 1/3) = 81 · (1 − 1/81)/(2/3) = 81 · (80/81) · (3/2) = 120.
Parameter Meaning Example
a₁ First term 1 in 1, 3, 9, …
r Common ratio 3 in 1, 3, 9, …
aₙ n‑th term a₅ = 16 in 1, 2, 4, 8, 16, …
Sₙ Sum of first n terms S₃ = 1+3+9 = 13

Fibonacci Sequence

The Fibonacci sequence starts with 0 and 1, and each new term is the sum of the previous two, creating a famous pattern in mathematics and nature.

Definition

F₀ = 0, F₁ = 1
Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2

Example 1: First few terms
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
Each term after the first two is the sum of the previous pair.
Example 2: Using the calculator
Set “Number of Terms” to 8.
Result: 0, 1, 1, 2, 3, 5, 8, 13 and sum = 33.
Index n Fibonacci Fₙ Comment
0 0 By definition
1 1 By definition
2 1 0+1
5 5 2+3
7 13 5+8

Real-World Applications of Sequences

Arithmetic, geometric, and Fibonacci sequences appear in finance, computing, biology, and many other areas where patterns and repetition matter.

💰 Finance and Savings

Regular deposits with constant increases can be modeled using arithmetic sequences, while compound interest uses geometric sequences.

📡 Signal and Data Processing

Geometric patterns describe exponential growth or decay in signals, error-correcting codes, and algorithm complexity.

🌱 Biology and Nature

Fibonacci numbers show up in branching patterns, leaf arrangements, and spirals in shells and sunflowers.

🎮 Computer Science

Sequences are used in loops, simulations, random number generators, and performance analysis involving geometric growth.

📊 Series and Calculus

Arithmetic and geometric series are foundational for understanding infinite series, convergence, and summation techniques.

🧮 Puzzles and Competitions

Recognizing sequence types quickly is valuable in math olympiads, aptitude tests, and coding challenges.

Common Mistakes and How to Avoid Them

Sequence problems often go wrong when the pattern is misidentified or formulas are misapplied, especially with indices and sums.

❌ MISTAKE 1: Mixing up d and r
Treating an arithmetic sequence as geometric (or vice versa) leads to incorrect terms and sums.
✅ Check whether the change is additive (difference) or multiplicative (ratio).
❌ MISTAKE 2: Using wrong index in formulas
Forgetting that aₙ = a₁ + (n−1)d and aₙ = a₁·rn−1 shifts results by one term.
✅ Always verify with small n (like n=1 or n=2) to confirm the formula.
❌ MISTAKE 3: Ignoring term limits
Very large n or extreme ratios produce huge numbers that are hard to interpret.
✅ Use reasonable term counts (like your calculator limits) and watch for overflow in sums.
❌ MISTAKE 4: Wrong Fibonacci starting values
Starting from 1,1 instead of 0,1 changes index alignment and sums.
✅ Confirm the definition: this calculator uses 0, 1, 1, 2, 3, … as the default Fibonacci sequence.

Practice Problems with Solutions

Use these problems to practice identifying sequence types, finding terms, and computing sums; then verify your answers with the calculators above.

Arithmetic Sequence
Problem 1: a₁ = 5, d = 4, find a₈ and S₈.
Show Solution

a₈ = 5 + 7·4 = 33.
S₈ = 8/2 · (5 + 33) = 4 · 38 = 152.

Problem 2: Sequence: 10, 7, 4, 1, … Find d and the 6th term.
Show Solution

d = −3 (each step subtract 3).
a₆ = 10 + 5·(−3) = 10 − 15 = −5.

Geometric Sequence
Problem 3: a₁ = 3, r = 2, find a₅ and S₅.
Show Solution

a₅ = 3·2⁴ = 3·16 = 48.
S₅ = 3·(1 − 2⁵)/(1 − 2) = 3·(1 − 32)/(−1) = 3·31 = 93.

Problem 4: Sequence: 81, 27, 9, 3, … Find r and S₄.
Show Solution

r = 1/3 (each term divided by 3).
S₄ = 81 + 27 + 9 + 3 = 120.

Fibonacci Sequence
Problem 5: List the first 7 Fibonacci numbers starting from F₀ = 0, F₁ = 1, then find their sum.
Show Solution

Sequence: 0, 1, 1, 2, 3, 5, 8.
Sum = 0+1+1+2+3+5+8 = 20.

Problem 6: A Fibonacci-like sequence starts with 2, 3 and each term is the sum of the previous two. Find the next 4 terms.
Show Solution

Terms: 2, 3, 5, 8, 13, 21.
Next four terms after 2, 3 are 5, 8, 13, 21.