# 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.*