How Do You Find the n-th Term in a Sequence?

Theory

Sequences

Numbers written in succession separated by a comma are called a sequence. They look like this:

a_{1}, a_{2}, a_{3}, a_{4}, \dots

The indices $n = 1, 2, 3, 4$ and so on tell you which term of the sequence an element is, making $a_{n}$ the $n$ th term of the sequence.

Example 1

$a_{3}$ is the third term of the sequence.

There’s often a pattern describing the sequence. This pattern helps you find next term of the sequence. Due to the existence of this pattern, you can create an expression that correlates an element and its place in the sequence.

To find this correlation, it can be useful to make a table with two rows. The top row shows which place the element is in the sequence, and the bottom row shows the element in that place. You can then create an expression for the sequence to find the terms that come after the ones you’ve seen. This can be done the hard way or the easy way. The hard way would be to find the expression through inspection or trial and error. The easy way would be to just use regression. Here’s an example:

Example 2

You have the numbers 2, 4, 6, 8 and 10, and want to find the next numbers of the sequence.

You can see that $2$ is the first term, $4$ is the second, $6$ is the third and so on. Using that information, you make this table:


Index, $n$	1	2	3	4	5

Term, $y$	2	4	6	8	10

You can see that the sequence increases by $2$ between each term. You can check this by picking a term at random and subtracting the term before it, like this:

8 - 6 = 2 .

You have to do this calculation for each term to be sure that it’s actually correct. As the difference between two consecutive terms is $2$ for the first five terms, you can assume that it will continue this way. You can see that the $y$ values are equal to $2$ times the index. That makes the expression for the $n$ th term of the sequence

y = 2 n .

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