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

## Nth Term Of Increasing Sequences Video

The n^{th} 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 n^{th} 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 (1^{st} row)

(2^{nd} row)

5 8 11 14 17 (3^{rd} 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 2^{nd} row).

n 1 2 3 4 5 (1^{st} row)

3n 3 6 9 12 15 (2^{nd} row)

5 8 11 14 17 (3^{rd} 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 3^{rd} row.

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

Therefore our n^{th} term = 3n + 2

**Example 2**

Find the n^{th} 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 (1^{st} row)

(2^{nd} row)

2 8 14 20 26 (3^{rd} row)

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

n 1 2 3 4 5 (1^{st} row)

6n 6 12 18 24 30 (2^{nd} row)

2 8 14 20 26 (3^{rd} row)

Now, to get the numbers in the 3^{rd} row from the 2^{nd} 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 n^{th} 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:

## Questions & Answers

- Helpful 2
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.

Helpful 3What 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.

Helpful 1What 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.

Helpful 14What is the nth term rule of the linear sequence below? 13 , 7 , 1 , − 5 , − 11

The sequence is going down by -6 so compare this sequence to -6, -12,,-18,-24, -30.

You will have to add on 19 to these negative multiples to give the numbers in the sequence.

Helpful 6

## Comments

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

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

good

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

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

i don't get it

i cant understand

this method really helps me :)

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

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.

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

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

14