Calculate the n-th term and the sum of an arithmetic or geometric sequence, and list the first terms. Enter your values; the result updates as you type, with the method shown below.
About this calculator
Arithmetic sequences add the same difference each time (2, 5, 8, 11…); geometric sequences multiply by the same ratio (2, 6, 18…). This tool gives the n-th term, the sum of the first n terms and lists the first terms.
Choose the type, then enter the first term, the difference or ratio, and n.
When to use it
- Maths homework.
- Savings that grow by a fixed amount or rate.
- Predicting a pattern.
- Teaching sequences.
How to read the result
Arithmetic: uₙ = u₁ + (n − 1)d, sum = n(u₁ + uₙ)/2. Geometric: uₙ = u₁ × rⁿ⁻¹, sum = u₁(rⁿ − 1)/(r − 1).
A geometric sequence with a ratio above 1 grows very fast.
Mistakes to avoid
- Using n instead of n − 1 in the formula.
- Mixing up difference and ratio.
- Starting the count at 0 when the course starts at 1.
Formula
aₙ = a₁ + (n − 1)d or aₙ = a₁ · r^(n − 1)
Worked example
Arithmetic, first term 2, difference 3, n = 10: 10th term 29, sum 155.
Frequently asked questions
Arithmetic or geometric?
Arithmetic adds the same number; geometric multiplies by the same number.
What is the sum formula?
Arithmetic: n(a₁ + aₙ)/2; geometric: a₁(1 − rⁿ)/(1 − r).
What is the difference between the two types?
Arithmetic adds, geometric multiplies.
What if the ratio is 1?
Every term is the same.
Does it start at term 1?
Yes, the first term is u₁.
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Results are estimates for information only. Check important figures with a professional.