Find the nth Term in Linear Sequences

An arithmetic or linear sequence is one like this:

 

 

We see that the common difference is 2.

So, we can say the rule starts with 2n

 

The problem is that the first number we would get by doing 2n is 2 x 1 which gives us 2 when we need to get 3!

Therefore, its simple - we just add 1

 

Answer = 2n + 1

 

The best thing about this is that it means we can find any term in that sequence.

 

We know the rule is 2n + 1

To find the 150th term simply substitute n for 150 (n is the term number)

Like this!         

2 x 150 + 1 = 301

 

If we want the 3,000th term we can substitute n for 3,000

Like this! 

2 x 3,000 + 1 = 6,001

How cool is that?

 

It also means we can generate the sequence from a rule given to us. How smart are we?

 

Let's take the rule 3n - 1 (generate the first five terms)

1st term will be when n = 1          3 x 1 - 1 = 2 (first term)

2nd term will be when n = 2         3 x 2 - 1 = 5 (second term) .......

 

We can save time here by cutting down the work!

 

We know the first term is 2 and the rule is add 3 so we can just add 3 to find the next ones.

 

2, (2 + 3 = 5), (5 + 3 = 8), (8 + 3 = 11) etc .......

2, 5, 8, 11, 14, ........

 

Let's do some questions to check you have got this!