About the Number Sequence Calculator
A number sequence calculator finds the nth term of an arithmetic or geometric sequence and the sum of its terms. These two patterns — adding a fixed amount each step, or multiplying by a fixed factor — describe an enormous range of real situations, from savings plans to compound growth.
The formula
Arithmetic: aₙ = a₁ + (n − 1)d Geometric: aₙ = a₁ · rⁿ⁻¹a₁ is the first term, n the term number, d the common difference added each step, and r the common ratio multiplied each step.
How to use this calculator
- 1Enter your First Term (a₁). The field starts at
2, which you can overwrite. - 2Enter your Common Difference / Ratio (d or r). The field starts at
3, which you can overwrite. - 3Enter your Find Term Number (n). The field starts at
10, which you can overwrite. - 4Enter your Type (1=Arithmetic, 2=Geometric). The field starts at
1, which you can overwrite. - 5Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| First Term (a₁) | 2 |
| Common Difference / Ratio (d or r) | 3 |
| Find Term Number (n) | 10 |
| Type (1=Arithmetic, 2=Geometric) | 1 |
Result
Sequence: 2, 5, 8, 11, 14...
10th Term: 29
Sum of 10 terms: 155
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
The distinction is additive versus multiplicative, and it produces radically different long-run behaviour. Arithmetic sequences grow in a straight line: 2, 5, 8, 11 adds 3 each time. Geometric sequences curve: 2, 6, 18, 54 multiplies by 3, and after twenty terms it is millions of times larger than the arithmetic equivalent.
This is the same distinction as simple versus compound interest, and it is why compounding dominates over long horizons. It also explains why a geometric sequence with a ratio between 0 and 1 converges — each term is a fraction of the last, so an infinite sum reaches a finite limit, which is what makes 0.999… equal exactly 1.
Things worth knowing
- Identify the type by checking differences and ratios. Constant difference means arithmetic; constant ratio means geometric.
- The sum of an arithmetic sequence is n times the average of the first and last terms.
- A geometric series with |r| < 1 converges to a₁/(1 − r) as n grows without limit.
- A negative ratio makes terms alternate in sign.
- The Fibonacci sequence is neither arithmetic nor geometric — each term is the sum of the previous two.
Frequently asked questions
How do I tell an arithmetic sequence from a geometric one?+
Check consecutive terms. If the difference is constant it is arithmetic; if the ratio is constant it is geometric. In 3, 7, 11, 15 the difference is always 4; in 3, 6, 12, 24 the ratio is always 2.
How do I find the nth term?+
For arithmetic, add (n − 1) differences to the first term. For geometric, multiply the first term by the ratio raised to (n − 1). The tenth term of 2, 5, 8 is 2 + 9 × 3 = 29.
Can an infinite series have a finite sum?+
Yes, if it is geometric with a ratio between −1 and 1. Each term shrinks, and the total converges to a₁/(1 − r). This is why 1/2 + 1/4 + 1/8 + … sums to exactly 1.
Where do geometric sequences appear in real life?+
Compound interest, population growth, radioactive decay, and depreciation all multiply by a constant factor each period, which is exactly a geometric sequence.