How to Find the nth Term of an Increasing Linear Sequence.

Updated on November 26, 2016

Nth Term Of Increasing Sequences Video

The nth term of a number sequence is a formula that gives you the values in the numbers sequence from the position number (some people call it the position to term rule).

Example 1

Find the nth term of this sequence.

5 8 11 14 17

First of all write the position numbers 1 to 5 above the top of the numbers in the sequence (call these numbers at the top n). Make sure you leave a gap.

n 1 2 3 4 5 (1st row)

(2nd row)

5 8 11 14 17 (3rd row)

Next, work out the difference between the terms in the sequence (also known as the term to term rule). It is quite clear that you are adding on 3 each time. This tells us that the nth term has something to do with the 3 times table. Therefore, you multiply all the numbers at the top by 3 (just write your multiples of 3). Do this in the space you have left (the 2nd row).

n 1 2 3 4 5 (1st row)

3n 3 6 9 12 15 (2nd row)

5 8 11 14 17 (3rd row)

Now, you can see that if you add on 2 to all the numbers on the second row you get the number in the sequence on the 3rd row.

So our rule is to times the numbers on the 1st row by 3 and add on 2.

Therefore our nth term = 3n + 2

Example 2

Find the nth term of this number sequence.

2 8 14 20 26

Again write the numbers 1 to 5 above the numbers in the sequence, and leave a spare line again.

n 1 2 3 4 5 (1st row)

(2nd row)

2 8 14 20 26 (3rd row)

Since the sequence is going up by 6, write down your multiples of 6 on the 2nd row.

n 1 2 3 4 5 (1st row)

6n 6 12 18 24 30 (2nd row)

2 8 14 20 26 (3rd row)

Now, to get the numbers in the 3rd row from the 2nd row take off 4.

So, to get from the position numbers (n) to the numbers in the sequence you have to times the position numbers by 6 and take off 4.

Therefore, the nth term = 6n – 4.

If you want to find the nth term of a number sequence using the nth term formula then check out this article:

How to find the nth term of an increasing linear sequence.

Questions & Answers

  • Which formula represents the nth term of the arithmetic sequence 2,5,8,11,....?

    The first differences are 3, so compare the sequence to the multiplies of 3 which are 3, 6, 9 ,12.

    You will then need to subtract 1 off these multiples of 3 to give the number in the sequence.

    So the final formula for this arithmetic sequence is 3n - 1.

  • What is the nth term rule of the linear sequence below? 2 , 5 , 8 , 11 , 14 , . . .

    The sequence is increasing by 3 each time so compare the sequence with the multiples of 3 (3,6,9,12,15...).

    You will then need to minus 1 from the multiples of 3 to give the numbers in the sequence.

    So the nth term is 3n - 1.

  • What is the nth term rule of the linear sequence below? − 5 , − 2 , 1 , 4 , 7

    The numbers are going up by 3 each time, so it has something to do with the multiples of 3 (3,6,9,12,15).

    You will need to take 8 off these multiples to give the numbers in the sequences.

    Therefore the nth term will be 3n - 8.

  • What is the middle term in -3,?, 9

    If the sequence is linear then it will be going up by the same amount each time.

    -3 + 9 is 6, and 6 divided by 2 is 3.

    So the middle term is 3.


Submit a Comment
  • profile image


    7 years ago

    What's value of a in this pattern 3a-4; 4a-3; 7a-6

  • profile image


    7 years ago

    THANK YOU WHOEVER YOU ARE!!! BECAUSE OF YOU I WILL NOT FAIL!!!! LOVE YOU!!!!!!!!! :D :D :D :D you teach even better than my teacher

  • profile image


    7 years ago


  • catman3000 profile imageAUTHOR


    7 years ago from England, UK

    Sheila, the nth term for your sequence in n^2. All you need to do is square the position number. 4^2 = 16, 5^2 = 25 and 50^ = 2500.

  • profile image


    7 years ago

    (1ST row) 1 2 3 4 5.....50

    (3rd row) 1 4 16 ( ), ( )......( )

  • profile image


    7 years ago

    i don't get it

  • profile image

    ramina kate 

    8 years ago

    i cant understand

  • profile image

    blah blah blah 

    8 years ago

    this method really helps me :)

  • profile image


    8 years ago

    I realised that for mine, this does not work as the 2nd row answer has similar pattern to the 3rd row. Here's my sequence: 2, 4, 7, 11, 16, 22

  • Shahid Bukhari profile image

    Shahid Bukhari 

    8 years ago from My Awareness in Being.

    You've found it my friend ... its indeed "unth" ... Because, linear calculations, begin, and end, with the Infinite infesting the origins and the end ... because, there are no numbers which could define the Infinite ... "unth" is as good a phonetic symbol, as any Greek symbol.

  • catman3000 profile imageAUTHOR


    8 years ago from England, UK

    Follow the method above if your sequence is linear. A linear sequence will be increasing or decreasing by the same amount each time.

  • profile image


    8 years ago

    i am trying to do maths homework on nth terms and my ks3 teacher has given me a patternin numbers for linear sequances. Do i just work it out the same way as normal numbers. thanx

  • catman3000 profile imageAUTHOR


    9 years ago from England, UK

    Yes it is possible but it will be a quadratic sequence you are looking at. I will publish an article soon on quadratic sequences.

  • profile image


    9 years ago

    what if you have a sequence which adds 1,2,3,4,5,6,7,8,9 etc. - is it possible to find thenth term, if so how?? thanx


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