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

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:

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