The nth term

In a nutshell

Sequences defined using an $n^{th}$ term formula help us find the sequence from the position of the terms in the sequence. The $n^{th}$ term method is more useful, as if we want to find the 100th term, it is not necessary to find any of the previous terms.

Generating a sequence

The $n^{th}$ term formula can be used to generate a sequence. Start by substituting $n=1$ into the formula to find the first term. Then substitute $n=2$ for the second term, $n=3$ for the third term and so on.

Examples

NTH TERM FORMULA	SEQUENCE
$n^{th} term = 4n+4$	$8,12, 16, 20,24$
$n^{th} term = 8-6n$	$2, -4, -10, -16, -22$
$n^{th} term = n^2+2$	$3, 6, 11, 18, 27$

Find the nth term formula for an arithmetic sequence

An arithmetic sequence is one where the differences between adjacent terms are always the same. The term-to-term rule would be to $+$ or $-$ the same number each time.

It is possible to find the $n^{th}$ term formula for an arithmetic sequence.

Example

Find the $n^{th}$ term formula for the sequence $2, 5, 8, 11$ .

Answer

Start by putting the sequence, and their positions $n$ into a table. Work out the difference each time, in this case the difference is $3$ , so add another row in the table for $3n$ . Then think about how to get from $3n$ to the sequence, in this case each term in our sequence is $1$ less than the $3n$ row, so the formula is $\underline{n^{th} \space term = 3n-1}$ .

$n$ (the position of each term)	$1$	$2$	$3$	$4$
$n^{th} \space term$ (the sequence)	$2$	$5$	$8$	$11$
the difference is $3$ so use $3n$	$3$	$6$	$9$	$12$
each term is $1$ less than $3n$ , so use $3n-1$	$2$	$5$	$8$	$11$

$\underline {n^{th} \space term = 3n-1}$

Note: Always check the formula, by substituting values for $n$ to see if it works.