Arithmetic and Geometric Sequence Calculator
Enter the first term and the common difference or ratio, and get the nth term, the sum of the first n terms, the first terms listed, and for a geometric sequence with a small enough ratio, the sum to infinity. You can also find the position of a value, or paste a list of numbers and let the tool say whether it is arithmetic or geometric and continue it. Everything runs in your browser.
2, 5, 8, 11, 14, 17, 20, 23, 26, 29
An arithmetic sequence adds the same difference each time, so term n is a1 + (n - 1)d and the first n terms add up to n / 2 times the first plus the last. A geometric sequence multiplies by the same ratio each time, so term n is a1 times r to the power (n - 1), and the first n terms add up to a1(1 - r^n) / (1 - r). When the ratio is between -1 and 1 the terms shrink and the sum to infinity is a1 / (1 - r). All of the arithmetic uses exact fractions, so 3/2 stays 3/2 and large terms keep every digit. Everything is worked out in your browser and nothing is uploaded.
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Math & MoneyAbout the Arithmetic and Geometric Sequence Calculator
An arithmetic sequence adds the same number, the common difference d, each time. Its nth term is a1 + (n - 1)d and the first n terms add up to n / 2 times the first plus the last. For 2, 5, 8 and so on, with d = 3, the 10th term is 29 and the first 10 terms add up to 155. A geometric sequence multiplies by the same number, the common ratio r. Its nth term is a1 times r to the power (n - 1), and the first n terms add up to a1(1 - r^n) / (1 - r), or n times a1 if r is 1. For 3, 6, 12 and so on, with r = 2, the 8th term is 384 and the first 8 terms add up to 765. When r is between -1 and 1 the terms shrink toward zero and the sum to infinity is a1 / (1 - r), so 1, 1/2, 1/4 and so on adds up to 2. All of the arithmetic is done with exact fractions, so a ratio of 3/2 stays 3/2, large terms keep every digit and nothing is rounded until it is shown. Numbers can be entered as decimals or fractions. The identify tab checks a list for a constant difference or a constant ratio, treats a list with a zero before the end as not geometric, and gives the next terms. Arithmetic sequences can have up to 100,000 terms and geometric ones up to 500. Nothing is uploaded.
How do I find the nth term of an arithmetic sequence?▾
Use a1 + (n - 1) x d. For the sequence 2, 5, 8, 11, the first term is 2 and d is 3, so the 10th term is 2 + 9 x 3 = 29.
How do I find the sum of an arithmetic sequence?▾
Multiply the number of terms by the average of the first and last term: S = n / 2 x (a1 + an). For the first 10 terms of 2, 5, 8, ..., that is 5 x (2 + 29) = 155.
How do I find the sum of a geometric sequence?▾
Use S = a1(1 - r^n) / (1 - r) when r is not 1. For 3, 6, 12, ... with 8 terms, S = 3 x (1 - 256) / (1 - 2) = 765. If r is 1, every term is the same and the sum is n x a1.
When does a geometric series have a sum to infinity?▾
When the ratio is between -1 and 1, because then the terms shrink toward zero. The sum is a1 / (1 - r). For 1, 1/2, 1/4, ... it is 2. If the ratio is 1 or more in size, the terms do not shrink and the sum does not settle.
How can I tell whether a list is arithmetic or geometric?▾
Subtract each term from the next: if all the differences are equal it is arithmetic. Divide each term by the one before: if all the ratios are equal it is geometric. The identify tab does both checks. Your numbers are not uploaded.