Number Sequence Calculator
Calculate arithmetic, geometric, and Fibonacci sequences with step-by-step solutions
Arithmetic Sequence Calculator
Formula: an = a₁ + f × (n-1)
Example: 1, 3, 5, 7, 9, 11, 13, ...
Geometric Sequence Calculator
Formula: an = a × r(n-1)
Example: 1, 2, 4, 8, 16, 32, 64, 128, ...
Fibonacci Sequence Calculator
Formula: a₀=0; a₁=1; an = an-1 + an-2
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
General Introduction to Sequences
A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in a sequence is called a term. The position of a term in the sequence is called its index.
Key Characteristics:
- Order matters: The sequence 1, 2, 3 is different from 3, 2, 1
- Length: A sequence can be finite (has an end) or infinite (continues forever)
- Pattern: Each sequence follows a specific rule that determines how terms are generated
Common Types of Sequences:
- Arithmetic Sequences: Each term is obtained by adding a constant difference to the previous term
- Geometric Sequences: Each term is obtained by multiplying the previous term by a constant ratio
- Fibonacci Sequence: Each term is the sum of the two preceding terms
Applications: Sequences are fundamental in mathematics, appearing in calculus (series), computer science (algorithms), economics (compound interest), and many other fields. Sequences can be convergent (approach a limit) or divergent (grow without bound).
Notation: Sequences are typically denoted as a1, a2, a3, ... or {a_n} where n is the index.
Arithmetic Sequence
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference and is denoted by f.
General Forms:
an = a₁ + f × (n-1)
Where: an = nth term, a₁ = first term, f = common difference, n = term number
an = am + f × (n-m)
Where: am = mth term (useful when you know a term other than the first)
Example: 1, 3, 5, 7, 9, 11, 13, ...
This is an arithmetic sequence with:
- First term (a₁) = 1
- Common difference (f) = 2
To find the 5th term: a₅ = 1 + 2 × (5-1) = 1 + 2 × 4 = 1 + 8 = 9 ✓
Sum of Arithmetic Sequence:
Sn = n × (a₁ + an) / 2
Where: Sn = sum of first n terms
Example: Sum of first 5 terms (1, 3, 5, 7, 9)
Direct calculation: 1 + 3 + 5 + 7 + 9 = 25
Using formula: S₅ = 5 × (1 + 9) / 2 = 5 × 10 / 2 = 50 / 2 = 25 ✓
Geometric Sequence
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by r.
General Form:
an = a × r(n-1)
Where: an = nth term, a = first term, r = common ratio, n = term number
Example: 1, 2, 4, 8, 16, 32, 64, 128, ...
This is a geometric sequence with:
- First term (a) = 1
- Common ratio (r) = 2
To find the 8th term: a₈ = 1 × 2(8-1) = 1 × 2⁷ = 1 × 128 = 128 ✓
Sum of Geometric Sequence:
Sn = a × (1 - rn) / (1 - r) (when r ≠ 1)
When r = 1: Sn = a × n
Example: Sum of first 3 terms (1, 2, 4)
Direct calculation: 1 + 2 + 4 = 7
Using formula: S₃ = 1 × (1 - 2³) / (1 - 2) = 1 × (1 - 8) / (-1) = 1 × (-7) / (-1) = 7 ✓
Special Cases:
- When |r| < 1, the sequence converges and the infinite sum exists: S∞ = a / (1 - r)
- When |r| ≥ 1, the sequence diverges (grows without bound)
Fibonacci Sequence
The Fibonacci sequence is a sequence where each number is the sum of the two preceding ones, typically starting with 0 and 1, or 1 and 1. The sequence is named after Leonardo of Pisa, known as Fibonacci.
Mathematical Form:
a₀ = 0; a₁ = 1; an = an-1 + an-2
Where: an = nth term (starting from index 0)
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Starting values: a₀ = 0, a₁ = 1
a₂ = a₁ + a₀ = 1 + 0 = 1
a₃ = a₂ + a₁ = 1 + 1 = 2
a₄ = a₃ + a₂ = 2 + 1 = 3
a₅ = a₄ + a₃ = 3 + 2 = 5
And so on...
Applications:
Computer Algorithms
The Fibonacci sequence appears in various algorithms, including dynamic programming problems, recursive function examples, and optimization techniques.
Economics
Used in financial modeling, particularly in the analysis of market trends and the Fibonacci retracement technique in technical analysis.
Biology
Appears in natural phenomena such as the arrangement of leaves on stems, the pattern of seeds in sunflowers, and the spiral patterns in shells.
Interesting Properties:
- The ratio of consecutive Fibonacci numbers approaches the golden ratio (φ ≈ 1.618) as n increases
- Every third Fibonacci number is even
- The sum of the first n Fibonacci numbers equals Fn+2 - 1
- Fibonacci numbers appear in Pascal's triangle