Arithmetic and Geometric Sequence Calculator
Find the nth term and the sum of an arithmetic or geometric sequence.
Two kinds of sequence
In an arithmetic sequence you add the same number each time: 3, 7, 11, 15… In a geometric sequence you multiply by the same number each time: 2, 6, 18, 54… The calculator finds any term without listing all the ones before it, and adds up the first n terms.
Example
On the first tab (arithmetic) with the starting values shown (First term 3; Common difference 4; Term number 10), The nth term comes out as 39. Other results include sum of the first n terms: 210.
Arithmetic sequences model steady growth such as a savings plan with fixed deposits. Geometric sequences model compound growth, and the sum to infinity exists only when the ratio is between −1 and 1.
Arithmetic sequences
Each term adds a constant difference d. The nth term is aₙ = a₁ + (n − 1) × d and the sum of the first n terms is Sₙ = n ÷ 2 × (a₁ + aₙ). For 3, 7, 11, 15, … (a₁ = 3, d = 4), the 10th term is 39 and the first 10 terms add up to 210. The famous sum 1 + 2 + … + 100 = 5,050 is an arithmetic series.
Geometric sequences
Each term multiplies by a constant ratio r. The nth term is aₙ = a₁ × rⁿ⁻¹ and the sum is Sₙ = a₁ × (rⁿ − 1) ÷ (r − 1). For 2, 6, 18, 54, … (a₁ = 2, r = 3), the 8th term is 4,374 and the first 8 terms add up to 6,560.
Infinite geometric series
When |r| is less than 1, the terms shrink and the sum converges to a₁ ÷ (1 − r). For 1 + 1/2 + 1/4 + 1/8 + …, the sum is 1 ÷ (1 − 0.5) = 2. If |r| is 1 or more, the series does not converge.
| Arithmetic | Geometric | |
|---|---|---|
| Change | Add d each time | Multiply by r each time |
| Graph | Straight line | Exponential curve |
| Example | 2, 5, 8, 11 | 2, 6, 18, 54 |
| Real-life | Saving a fixed amount | Compound interest |
Next steps: Exponent and Root Calculator, Compound Interest Calculator, Statistics Calculator.
Frequently asked questions
How do I find the nth term of an arithmetic sequence?
Use a₁ + (n − 1) × d.
How do I find the common ratio?
Divide any term by the term before it.
What is the sum of the first 100 numbers?
5,050.
When does a geometric series have a sum to infinity?
When the common ratio is between −1 and 1.
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