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Maths

The nth term

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The nth term

In a nutshell

Sequences defined using an nthn^{th} term formula help us find the sequence from the position of the terms in the sequence. The nthn^{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 ​nthn^{th} term formula can be used to generate a sequence. Start by substituting n=1n=1 into the formula to find the first term. Then substitute n=2n=2 for the second term, n=3n=3 for the third term and so on.​


Examples

NTH TERM FORMULA

SEQUENCE

nthterm=4n+4n^{th} term = 4n+4​​
8,12,16,20,248,12, 16, 20,24​​
nthterm=86nn^{th} term = 8-6n​​
2,4,10,16,222, -4, -10, -16, -22​​
nthterm=n2+2n^{th} term = n^2+2​​
3,6,11,18,273, 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 nthn^{th} term formula for an arithmetic sequence.


Example

Find ​the nthn^{th} term formula for the sequence 2,5,8,112, 5, 8, 11.

Answer

Start by putting the sequence, and their positions nn into a table. Work out the difference each time, in this case the difference is 33, so add another row in the table for 3n3n. Then think about how to get from 3n3n to the sequence, in this case each term in our sequence is 11 less than the 3n3n row, so the formula is nth term=3n1\underline{n^{th} \space term = 3n-1}.


nn (the position of each term)
11​​
22​​
33​​
44​​
nth termn^{th} \space term (the sequence)
22​​
55​​
88​​
1111​​
the difference is 33​ so use 3n3n​​
33​​
66​​
99​​
1212​​
each term is 11 less than 3n3n, so use 3n13n-1​​
22​​
55​​
88​​
1111​​


nth term=3n1\underline {n^{th} \space term = 3n-1}


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


Frequently Asked Questions (FAQ)

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

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