How to use the Arithmetic and Geometric Sequence Calculator
- Choose arithmetic or geometric.
- Enter the first term and the common difference or ratio.
- Enter how many terms you want to include.
- Read the nth term, the sum, the first terms and the infinite sum where it applies.
Formula used
Arithmetic: aₙ = a₁ + (n − 1)d and Sₙ = n ÷ 2 × (a₁ + aₙ). Geometric: aₙ = a₁ × rⁿ⁻¹ and Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r), or n × a₁ if r = 1. The infinite sum is a₁ ÷ (1 − r) when |r| < 1.
Worked example
Here is the Arithmetic and Geometric Sequence Calculator worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Type | Arithmetic (add a common difference) |
| First term (a₁) | 3 |
| Common difference or ratio | 4 |
| Number of terms (n) | 10 |
| Result | |
| nth term | 39 |
| Sum of the first n terms | 210 |
| First terms | 3, 7, 11, 15, 19, 23, 27, 31, … |
| Sum to infinity | Not applicable |
Arithmetic sequences
The nth term is aₙ = a₁ + (n − 1)d. Starting at 3 and adding 4 each time gives 3, 7, 11, 15, 19, 23, 27, 31, …. The 10th term is 3 + 9 × 4 = 39. The sum of the first n terms is n ÷ 2 × (first + last) = 10 ÷ 2 × (3 + 39) = 210.
The famous example is adding the whole numbers from 1 to 100. With a₁ = 1, d = 1 and n = 100, the sum is 5,050, the result attributed to the young Gauss.
Geometric sequences
The nth term is aₙ = a₁ × rⁿ⁻¹. Starting at 2 and multiplying by 3 for 6 terms, the last term is 486 and the terms are 2, 6, 18, 54, 162, 486. The sum of the first n terms is a₁ × (1 − rⁿ) ÷ (1 − r) = 728. If the ratio is 1, every term is equal and the sum is n × a₁.
Geometric growth is what compound interest follows: a balance multiplied by the same factor every period.
Infinite sums
A geometric series can add up to a finite number if the terms shrink, which needs a ratio between −1 and 1. Starting at 8 and halving each time, the terms 8, 4, 2, 1, … add up to 16, because the sum to infinity is a₁ ÷ (1 − r) = 8 ÷ 0.5.
If the ratio is 1 or more in size, the terms do not shrink, and the series has no finite sum. Arithmetic sequences with a non-zero difference never have a finite infinite sum.
Uses
Arithmetic sequences model steady changes: a salary that rises by a fixed amount every year, evenly spaced seats or savings of a fixed amount each month. Geometric sequences model growth by a percentage, repeated halving, bouncing balls and depreciation.
Loan and savings formulas, such as the annuity and SIP calculators, are sums of geometric series.
Frequently asked questions
What is the difference between arithmetic and geometric?
Arithmetic adds a constant each step. Geometric multiplies by a constant each step.
How do I find the nth term?
Arithmetic: a₁ + (n − 1)d. Geometric: a₁ × rⁿ⁻¹.
When does a geometric series converge?
When the common ratio r satisfies −1 < r < 1.
Can the difference or ratio be negative or a decimal?
Yes. A negative ratio makes the terms alternate in sign.
Why is the number of terms limited to 1000?
To keep results within the range of ordinary numbers. Very large powers can overflow.
What is the sum formula for arithmetic sequences?
n ÷ 2 × (first term + last term).
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.