Finding the rule from the terms
Look at the gaps. If every gap is the same, the sequence is arithmetic and the gap is the common difference.
For 3, 7, 11, 15 the gaps are all 4, so the rule is 4n − 1: put n = 1 and you get 3.
Worked example
The sequence 3, 7, 11, 15 has a common difference of 4 and a rule of 4n − 1.
The 10th term is 39 and the sum of the first ten terms is 210.
Geometric sequences
If each term is the same multiple of the one before, the sequence is geometric and the rule is a × r to the power n − 1.
For 2, 6, 18, 54 the ratio is 3, so the rule is 2 × 3^(n − 1).
Using the rule
The rule lets you jump straight to any term without listing the ones in between, which is the whole point of it.
It also tells you the shape of the sequence: an arithmetic rule adds the same amount each time, a geometric one multiplies.
Inputs for the nth term calculator
What do you have?
A list of terms to read, or a first term and a common difference.
Terms of the sequence
At least three terms, separated by commas or spaces, so the pattern can be checked.
Position
Which term you want. Position 1 is the first term.
CALCULATOR QUESTIONS
Nth term calculator questions
How do I find the nth term of 3, 7, 11, 15?
The gaps are all 4, so the rule is 4n − 1. Check it with n = 1: 4 − 1 = 3.
What is the 10th term of 3, 7, 11, 15?
39, because 4 × 10 − 1 = 39.
What if the gaps are not the same?
Then it may be geometric, where each term is a fixed multiple of the one before. If neither pattern holds, the sequence has no single nth term rule.
What is the sum of the first n terms?
For an arithmetic sequence it is n/2 × (2a + (n − 1)d). The calculator works it out from the same rule.