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# The nth term 0%

Summary

# 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 $2, 5, 8, 11$.

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.

FAQs

• Question: How do you find the nth term formula for an arithmetic sequence?

Answer: Start by finding the difference between term, this is the number that goes with n. For example, if the difference is +3, then use 3n. Then either add or subtract a constant until it matches the sequence.

• Question: Is nth term better than term-to-term rule?

Answer: The nth term method of defining a sequence can be more useful than the term-to-term rule, as it does not rely on finding any other previous terms in the sequence.

• Question: What is an nth term formula?

Answer: An nth term formula is a way of defining a sequence. The 'n' stands for the position of the term in the sequence. We can substitute different values of n to find the term.

Theory

Exercises