Arithmetic and Geometric Sequence Calculator
A sequence is a list of numbers that follows a rule. In an arithmetic sequence you add the same amount each time. In a geometric sequence you multiply by the same amount each time. Pick the type and enter where it starts.
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How it works
Arithmetic: the nth term is the first term plus (n minus 1) times the common difference. The sum of the first n terms is n times the first term plus the last term, divided by 2.
Geometric: the nth term is the first term times the ratio raised to the power n minus 1. The sum is the first term times (the ratio to the power n, minus 1), divided by (the ratio minus 1). If the ratio is 1, every term is the same, so the sum is the first term times n.
Worked example
The sequence starts at 2 with a step or ratio of 3. Term number 10 is 29, and the first 10 terms add up to 155.
Questions
What is the difference between a step and a ratio?
A step is added each time, as in 2, 5, 8. A ratio is multiplied each time, as in 2, 6, 18. Choose the type above to match your sequence.
Can the ratio be zero or negative?
Yes. A ratio of minus 2 makes the terms alternate in sign, which is still a valid sequence.
Why does the sum use a formula?
Adding up many terms one at a time is slow. The formula gives the same total in one step.
More like this
- Geometric Mean CalculatorAverage for growth rates and ratios.
- Percent Change CalculatorPercent up or down between two numbers.
- Average CalculatorMean of a list of numbers.
- Percent Difference CalculatorGap between two values, as a share of their average.
- Standard Deviation CalculatorSpread of a list of numbers.
- Weighted Average CalculatorAverage where some values count more.