Number Sequence Calculator
Decode mathematical patterns and project series growth instantly.
Number Sequence Calculator
Number sequences follow patterns that can be described by formulas. This calculator identifies arithmetic and geometric sequences, finds any term, and calculates the sum of any range of terms.
How to Use This Calculator
- Enter the sequence type: arithmetic or geometric.
- Enter the first term (a₁) and either the common difference (d) or common ratio (r).
- Enter the term number (n) you want to find.
- Optionally enter a range to calculate the sum of terms.
- Click Calculate to see the nth term and partial sum.
Key Sequence Formulas
Arithmetic sequence: aₙ = a₁ + (n−1)d
Sum of n terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d]
Geometric sequence: aₙ = a₁ × r^(n−1)
Sum of n terms: Sₙ = a₁ × (1−rⁿ)/(1−r) for r ≠ 1
Infinite geometric series: S = a₁/(1−r) for |r| < 1
Example Calculations
Arithmetic: First term = 3, difference = 4. Sequence: 3, 7, 11, 15, 19...
10th term: 3 + (9×4) = 39 | Sum of first 10: 10/2 × (3+39) = 210
Geometric: First term = 2, ratio = 3. Sequence: 2, 6, 18, 54...
6th term: 2 × 3⁵ = 486
Common Sequences in Nature and Math
- Fibonacci: 1, 1, 2, 3, 5, 8, 13... (each term = sum of previous two)
- Triangular numbers: 1, 3, 6, 10, 15... (Sₙ = n(n+1)/2)
- Square numbers: 1, 4, 9, 16, 25... (aₙ = n²)
- Prime numbers: 2, 3, 5, 7, 11, 13... (no simple closed formula)
Common Mistakes to Avoid
- Confusing arithmetic and geometric sequences — Arithmetic: constant difference (add/subtract). Geometric: constant ratio (multiply/divide). Check which applies before using a formula.
- Using n = 0 as the first term — Most sequence formulas use n = 1 for the first term. Confirm the indexing convention before plugging in.
- Applying geometric sum formula when |r| ≥ 1 for infinite series — The infinite geometric series only converges (has a finite sum) when |r| < 1.
Frequently Asked Questions
What is the Fibonacci sequence used for?
The Fibonacci sequence appears in plant growth patterns (spiral arrangements of seeds, leaves, and petals), stock market analysis (Fibonacci retracement levels), computer algorithms, and the golden ratio (consecutive Fibonacci terms approach φ ≈ 1.618).
What is an arithmetic series vs. arithmetic sequence?
A sequence is a list of terms. A series is the sum of terms in a sequence. Arithmetic sequence: 2, 5, 8, 11. Arithmetic series: 2 + 5 + 8 + 11 = 26.
Conclusion
Number sequences are everywhere — from nature to finance to computer science. Use this calculator to find any term or partial sum in arithmetic and geometric sequences, and explore the mathematical patterns that shape our world.
Related: Statistics Calculator | Probability Calculator | Percent Calculator
Application Note
Arithmetic sequences are great for simple growth, while Geometric sequences model compounding effects like investment returns or population expansion.
