About this Number Sequence
Number sequences are lists of numbers that follow a specific pattern. The two most common types are Arithmetic (adding a constant) and Geometric (multiplying by a constant). This tool solves for any term in the sequence or the total sum.
Who this calculator is for
- Students studying arithmetic and geometric sequences in algebra.
- Teachers and tutors checking nth-term or series-sum homework.
- Anyone modeling repeated change, growth, or step-by-step patterns.
When to use it
- Use this calculator to find the nth term of an arithmetic or geometric sequence.
- Use it to calculate the sum of the first n terms.
- Use it when you need to compare additive growth with multiplicative growth.
How to use this calculator
- Select the Type: Arithmetic or Geometric.
- Enter the First Term (a₁).
- Enter the Common Difference (d) or Ratio (r).
- Enter the Term Number (n) you want to find.
Key assumptions
- The term number n should be a positive whole number.
- Arithmetic mode uses an additive difference; geometric mode uses a multiplier.
- The calculator returns the finite sum for the first n terms, not an infinite-series result.
Accuracy and testing
Review formula sources, default test cases, assumptions, unit handling, rounding rules, and changelog for this calculator.
How this calculator works
Formulas used:
Arithmetic:
* Nth Term (aₙ) = a₁ + (n-1)d
* Sum (Sₙ) = (n/2) × (2a₁ + (n-1)d)
Geometric:
* Nth Term (aₙ) = a₁ × r^(n-1)
* Sum (Sₙ) = a₁ × (1 - r^n) / (1 - r)
Inputs explained
First Term (a1)
The starting number of the sequence.
Common Diff/Ratio
For Arithmetic, the number added each time. For Geometric, the multiplier.
Term Number (n)
The position in the sequence you want to calculate.
Understanding your results
Nth Term is the value at position n. Sum is the total of all numbers from position 1 to n.
Example scenarios
Arithmetic
Start 2, Diff 3. Sequence: 2, 5, 8, 11... 10th Term = 29. Sum = 155.
Geometric
Start 3, Ratio 2. Sequence: 3, 6, 12, 24... 10th Term = 1536. Sum = 3069.
Key Features
- Handles Arithmetic & Geometric
- Calculates Sum (Sₙ)
- Calculates Term (aₙ)
Tips & Best Practices
- Geometric sequences grow extremely fast (exponentially).
- If the ratio is between -1 and 1, a geometric series converges to a finite sum as n approaches infinity.
Glossary of Terms
- Arithmetic Progression
- A sequence where the difference between consecutive terms is constant.
- Geometric Series
- A sequence where each term is found by multiplying the previous term by a fixed, non-zero number.