Number Sequence Calculator
With first term 3, common difference or ratio 5, which term 10, number sequence comes to 48.000000 — the nth term. It is reached in 7 steps, the last of which is 3 + (10 - 1) * 5, and each one is printed on the page with its numbers filled in. The formula is the one published by Wolfram MathWorld, not an approximation fitted to it.
The nth term and the running total of an arithmetic or geometric sequence, and whether an infinite geometric one converges.
Formula and sources checked · How we check
First term 3, Sequence Arithmetic — add each time, Common difference or ratio 5, Which term 10
48.000000
The nth term for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- The nth term
3 + (10 - 1) * 548- Sum of the first n terms
10 * (3 + 48) / 2255- Average term
255 / 1025.5- An infinite sum converges
0- Sum to infinity, where it converges
0- Second term
3 + 58- Third term
3 + 2 * 513
Worked example
Starting at 3 and adding 5: the tenth term is 48 and the first ten add to 255. That sum is ten times the average of the first and last term, which is the trick Gauss is said to have found as a schoolboy asked to add the numbers to a hundred.
How to work it out yourself
- 1.Pick the kind first. A sequence that adds is arithmetic; one that multiplies is geometric, and the same word "common" is used for both the difference and the ratio.
- 2.The nth term formula uses n − 1, not n, because the first term has had the step applied zero times. That off-by-one is the commonest error in the topic.
- 3.For a geometric sequence, check the ratio: below one in absolute value the terms shrink and an infinite sum has a finite answer.
The formula
- The nth term
3 + (10 - 1) * 5 - Sum of the first n terms
10 * (3 + 48) / 2 - Average term
255 / 10 - An infinite sum converges
- Sum to infinity, where it converges
- Second term
3 + 5 - Third term
3 + 2 * 5
Source: Wolfram MathWorld — arithmetic series, Wolfram MathWorld — geometric series
Questions people actually ask
- What is the formula for the nth term?
- For an arithmetic sequence, a + (n − 1)d — the first term plus the step taken n − 1 times. For a geometric one, a × r^(n−1). Both use n − 1 because the first term has had nothing done to it yet, and using n instead shifts the whole sequence by one place.
- How do you sum a long arithmetic sequence quickly?
- Multiply the number of terms by the average of the first and the last. Pairing the ends of the list gives the same total every time — 1 + 100, 2 + 99 — which is why the numbers to a hundred sum to 5,050 without adding anything a hundred times.
- When does an infinite geometric series have a finite sum?
- When the ratio is between −1 and 1, so the terms shrink toward zero fast enough. Then the sum is the first term over one minus the ratio: 1 + ½ + ¼ + … = 2. Outside that range the terms do not shrink and the sum grows without limit.
- Is the Fibonacci sequence arithmetic or geometric?
- Neither. Each term is the sum of the two before it, which is a recurrence rather than a fixed step or ratio — and no closed nth-term formula of this shape describes it. Its ratio of consecutive terms does approach the golden ratio, 1.618, which is why it looks geometric from a distance.
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