How to use the arithmetic and geometric sequence calculator
- Pick the sequence type and the task: find the terms, find the common difference or ratio from the two ends, or find which term equals a value.
- Enter the first term and the values the page asks for, such as the common difference, the common ratio, the number of terms or the last term.
- Read the answer in the headline, and the sum and the other results in the list below it.
- For the terms task, look at the table of the first terms with their running sum.
Formula
Arithmetic: an = a1 + (n - 1) d, and Sn = n (a1 + an) / 2
Geometric: an = a1 x r^(n - 1), and Sn = a1 (r^n - 1) / (r - 1), or n x a1 when r = 1
Infinite geometric sum = a1 / (1 - r) when the size of r is less than 1
Difference from two ends d = (an - a1) / (n - 1); ratio r = (an / a1)^(1 / (n - 1))
An arithmetic sequence adds the same number each step and a geometric sequence multiplies by the same number. The ratio can be negative, and a ratio of size less than 1 makes the terms shrink toward zero.
Two sequences worked by hand
The arithmetic sequence that starts at 3 and adds 4 each time has a10 = 3 + 9 x 4 = 39 and a sum of 10 x (3 + 39) / 2 = 210. The number 99 is term 25, since (99 - 3) / 4 + 1 = 25, and the first 25 terms add to 1,275. Going from 5 to 95 in 10 terms needs a difference of 90 / 9 = 10 and the sum is 500.
The geometric sequence that starts at 2 and multiplies by 3 has a5 = 2 x 81 = 162 and a sum of 2 x (243 - 1) / 2 = 242. Starting at 100 with a ratio of 0.5 for 8 terms gives a last term of 0.78125 and a sum of 199.21875, and with infinitely many terms the sum is 200. Going from 3 to 768 in 5 terms needs a ratio of 4, or -4, since the number of steps is even.
Choosing the task
Use the terms task when you know the start and the step. Use the difference or ratio task when you know the first and the last term and the number of terms, such as saving from $5 to $95 in 10 payments. Use the term count task when you want to know whether a value belongs to the sequence and at which position.
If you only have a list of numbers and want to know the rule, the sequence pattern finder looks for it. The Fibonacci calculator covers the best known recursive sequence, and the geometric mean calculator finds the middle term of a geometric run.
Where the limits are
The page works with up to 1,000 terms and stops when a term or a sum passes 1e15, so a ratio such as 10 over 100 terms is refused instead of showing a rounded result. A negative ratio gives terms that alternate in sign. When the ratio is 1 or the difference is 0 every term is equal, and the term count task says so.
Assumptions and limits
- Terms are counted from 1, so a1 is the first term and an is the term in position n.
- The number of terms is a whole number from 1 to 1,000.
- For a ratio from the two ends only real ratios are found, and a negative one is listed when it also works.
- Numbers are shown with up to 10 significant digits.
Frequently asked questions
How do I find the nth term of an arithmetic sequence?
Use an = a1 + (n - 1) d. For a first term of 3 and a difference of 4, the 10th term is 3 + 9 x 4 = 39.
How do I find the sum of a geometric series?
Use Sn = a1 (r^n - 1) / (r - 1). For a1 = 2, r = 3 and n = 5 the sum is 2 x 242 / 2 = 242. If r is 1 the sum is n x a1.
When does a geometric series have an infinite sum?
When the size of the ratio is less than 1. Then the sum of all terms is a1 / (1 - r), so 100 with a ratio of 0.5 adds up to 200.
How do I know whether a number is in the sequence?
Use the term count task. For an arithmetic sequence the position (an - a1) / d + 1 must be a whole number of at least 1, and the page checks it for you.