Math · Sequences & Series
Number Sequence
Calculator
Generate terms, find the nth term, and calculate partial sums for arithmetic, geometric, Fibonacci, square, cube, and prime sequences. Includes sequence identifier.
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Nth Term
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Partial Sum
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Type
Arithmetic Sequence
aₙ = a₁ + (n−1) × d
Sum: Sₙ = n/2 × (2a₁ + (n−1)d) = n/2 × (a₁ + aₙ)
a₁
d
n
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Geometric Sequence
aₙ = a₁ × r^(n−1)
Sum: Sₙ = a₁ × (rⁿ − 1) / (r − 1) [r ≠ 1]
Infinite sum (|r|<1): S∞ = a₁ / (1 − r)
a₁
r
n
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Fibonacci Sequence
F(n) = F(n−1) + F(n−2), with F(1)=1, F(2)=1
Closed form: F(n) = (φⁿ − ψⁿ) / √5 where φ = (1+√5)/2 ≈ 1.618
Custom: set your own starting values below.
F₁
F₂
n
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Square Numbers (Perfect Squares)
aₙ = n²
1, 4, 9, 16, 25, 36, 49, 64, 81, 100…
Sum: Sₙ = n(n+1)(2n+1) / 6
n
qty
Cube Numbers (Perfect Cubes)
aₙ = n³
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000…
Sum: Sₙ = [n(n+1)/2]² = (Sₙ of natural numbers)²
n
qty
Prime Number Sequence
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…
No closed-form formula — generated by Sieve of Eratosthenes.
The nth prime grows approximately as: pₙ ≈ n × ln(n)
n
qty
Sequence
Terms
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Types of Number Sequences
A number sequence is an ordered list of numbers following a rule or pattern. Understanding sequences is fundamental to algebra, calculus, computer science, and natural phenomena like population growth and musical harmony.
Key Sequence Formulas
Arithmetic: aₙ = a₁ + (n−1)d Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁ × rⁿ⁻¹ Sₙ = a₁(rⁿ−1)/(r−1)
Fibonacci: F(n) = F(n−1)+F(n−2) F(1)=F(2)=1
Square: aₙ = n² Sₙ = n(n+1)(2n+1)/6
Cube: aₙ = n³ Sₙ = [n(n+1)/2]²
Triangular: aₙ = n(n+1)/2 1, 3, 6, 10, 15, 21…
How do you find the common difference of an arithmetic sequence?
Subtract any term from the next: d = aₙ₊₁ − aₙ. For example, in 3, 7, 11, 15, 19: d = 7−3 = 4. Check: 11−7=4, 15−11=4 ✓. If differences aren't constant, it's not arithmetic. For finding a₁ and d from two known terms: d = (aₘ − aₙ)/(m−n), then a₁ = aₙ − (n−1)d.
What is the Golden Ratio and why does it appear in Fibonacci?
The Golden Ratio φ = (1+√5)/2 ≈ 1.618034... As you take consecutive Fibonacci numbers and divide, the ratio approaches φ: 1/1=1, 2/1=2, 3/2=1.5, 5/3=1.667, 8/5=1.6, 13/8=1.625, ... → 1.618... The closed-form Binet formula for Fibonacci numbers is F(n) = (φⁿ − ψⁿ)/√5 where ψ=(1−√5)/2 ≈ −0.618.
When does a geometric series converge to a finite sum?
An infinite geometric series Σ a₁rⁿ converges (has a finite sum) only when |r| < 1. The sum is S∞ = a₁/(1−r). For example: 1 + 1/2 + 1/4 + 1/8 + ... = 1/(1−1/2) = 2. If |r| ≥ 1, the series diverges. This is why Zeno's paradox (each step is half the previous) resolves to a finite total: the infinite sum converges.
What is the sum of the first n natural numbers?
The sum 1+2+3+...+n = n(n+1)/2. This is the triangular number formula, famously derived by Gauss as a child when asked to sum 1 to 100: he paired 1+100=101, 2+99=101, ..., 50+51=101 — fifty pairs of 101 = 5050. The general formula: Sₙ = n(n+1)/2. For n=100: 100×101/2 = 5050 ✓.