# How to Find the Nth Term of an Increasing Linear Sequence

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Mark, a math enthusiast, loves writing tutorials for stumped students and those who need to brush up on their math skills.

## Nth Term of Increasing Sequences

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.

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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

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

Answer: 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.

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

Answer: 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.

Question: What is the nth term rule of the linear sequence below? 13,7,1,-5,-11

Answer: This is a decreasing sequence, -6n + 19.

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

Answer: 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.

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

Answer: 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.

Question: What is the nth term for the sequence 7,9,11,13,15?

Answer: Its going up in twos so the first term is 2n.

Then add on five to the multiples of 2 to give 2n + 5.

Question: What is the middle term in -3,?, 9

Answer: 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.

Sedzani on February 12, 2012:

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

algebra.. on November 03, 2011:

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

shit on October 30, 2011:

good

Mark (author) from England, UK on September 27, 2011:

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.

sheila on September 26, 2011:

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

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

jimmy on September 25, 2011:

i don't get it

ramina kate on August 19, 2011:

i cant understand

blah blah blah on March 31, 2011:

this method really helps me :)

regine on March 17, 2011:

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 from My Awareness in Being. on December 12, 2010:

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.

Mark (author) from England, UK on September 09, 2010:

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

boy on September 09, 2010:

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

Mark (author) from England, UK on June 30, 2010:

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

Girl on May 04, 2010:

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