How to use it
- Choose the Type of sequence: arithmetic or geometric.
- Choose What to find. Then type the First term a₁ and the Common difference d (arithmetic) or common ratio r (geometric), or give two known terms.
- Type the Term number or number of terms, n where the boxes ask for it.
- Read Answer and the First ten terms table, and go through How it is worked out for each step.
Arithmetic and geometric sequences
An arithmetic sequence adds the same difference d to each term, so aₙ = a₁ + (n − 1)d. A geometric sequence multiplies by the same ratio r each time, so aₙ = a₁ rⁿ⁻¹. The sum of the first n terms is Sₙ = n(a₁ + aₙ) ÷ 2 for an arithmetic sequence, and Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for a geometric one.
arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ) ÷ 2
geometric: aₙ = a₁ rⁿ⁻¹ and Sₙ = a₁(1 − rⁿ) ÷ (1 − r)- Two terms in a row give the difference d = a₂ − a₁, or the ratio r = a₂ ÷ a₁ when a₁ is not 0.
- A geometric sequence whose first term is 0 is 0 everywhere, so its ratio cannot be read from its terms.
- A negative ratio makes the terms alternate in sign, and the same formulas apply.
Infinite sums and what the calculator does not cover
A geometric series has a finite sum only when |r| < 1. Then the terms shrink and the sum is S∞ = a₁ ÷ (1 − r). When |r| is 1 or more, the partial sums do not settle, and the page says so.
- An arithmetic series never has a finite infinite sum, so that case is reported as having no finite sum.
- Term numbers above 10000 are refused, so that the working stays short.
- A ratio found from two terms that is not an exact root is given as a decimal approximation.
Worked example: 2, 5, 8, … and the 10th term
The first term is a₁ = 2 and the difference is d = 3, so a₁₀ = 2 + (10 − 1) × 3 = 29. The sum of the first ten terms is S₁₀ = 10 × (2 + 29) ÷ 2 = 155.
The geometric sequence 2, 6, 18, … has a₁₀ = 2 × 3⁹ = 39,366 and S₁₀ = 59,048. The series 2 + 1 + 1/2 + … has ratio r = 1/2, so its infinite sum is S∞ = 2 ÷ (1 − 1/2) = 4.
| Question | Answer |
|---|---|
| 10th term, arithmetic (2, 5, 8, …) | 29 |
| Sum of 10 terms, arithmetic | 155 |
| 10th term, geometric (2, 6, 18, …) | 39,366 |
| Sum of 10 terms, geometric | 59,048 |
| Infinite sum of 2 + 1 + 1/2 + … | 4 |
Frequently asked questions
How do I find the common difference of an arithmetic sequence?
Subtract a term from the next one: d = a₂ − a₁. From two terms with known positions, use the task d or r from two known terms: d = (t₂ − t₁) ÷ (k₂ − k₁).
When does an infinite geometric series have a sum?
Only when the absolute value of the ratio is less than 1. Then the terms get smaller and the sum is S∞ = a₁ ÷ (1 − r). With a₁ = 2 and r = 0.5 the sum is 4.
Why does the page show 2 × 3⁹ and not only the decimal?
The formula is shown with its numbers so that you can follow it. The exact answer is written in full whenever it is short, and a very long answer appears in scientific form with its digit count.
Can the common ratio be negative?
Yes. A ratio such as −2 makes the terms alternate in sign, and the sum still follows the same formula. An even number of steps between two known terms also fits a negative ratio of the same size.
Is the calculation done on a server?
No. The sequence is computed in your browser, and nothing you enter is sent to us.
Sources
Figures and text last checked on Tuesday, November 10, 2026.