When I started looking at “Sequences and Proof”, I fell into the trap of looking at some of the most difficult questions (A and A*) first. What a surprise to find that I couldn’t do them! I now think it’s especially important with this topic, to master the basic, simple stuff and then move onto more challenging areas such as proof using algebra.

So, the basics:-

**A sequence is a list of numbers in a given order**

**Consecutive numbers are numbers next to each other**

**We call the numbers in a sequence, terms**

**
**This leads us to the most basic sequence:-

**A liner sequence increases or decreases in equal sized steps.**

**
**For example:- 2, 4, 6, 8

In this sequence the terms increase by 2 in each step.

We can say: The nth term = 2n (i.e the 1^{st} term = 1 x 2 = 2, the 2^{nd} term = 2 x 2 = 4, the 3^{rd} term = 3 x 2 = 6 and the 4^{th} term = 4 x 2 = 8).

When you’re trying to understand a sequence it’s very useful to write the differences between each term under the sequence, like this:-

When you have a linear sequence if you look at differences between the terms to start to define the sequence, if the differences are all 2 then the sequence definition will include 2n, if the differences are all 3 then the sequence definition will include the term 3n, if the differences are all 4 then the the sequence definition will include the term 4n etc., etc.

This is best explained with a couple of examples:-

Find the nth term of the sequence 4, 6, 8, 10,.…

First look for the difference between each term:-

You now know that the sequence includes 2n. Set the information in a table including 2n:-

This method might seem over the top for simple linear sequences, but it shows its worth with more complicated examples:-

Find the nth term of the sequence 1, 9, 17, 25

So start by finding the difference between each term and then complete the grid as shown above:-

Most questions ask you to find the nth term but sometimes this is turned on its head. For example what is the 10^{th} term if the nth term = 8n — 7? Just feed the actual number into the statement:

**10 ^{th} term = (8 x 10) — 7 = 73 **

Alternatively you might be asked to find out if a particular number is a term in the sequence, for example is 155 a term in the above sequence? Again just feed the number into the sequence definition.

If 155 is in the sequence, then 8n — 7 = 155

and 8n = 162

n = 162/8 = 20.25

So “n” is not a whole number and 155 is not in the sequence.

Another example using the same sequence, is 177 a term in the sequence?

If 177 is in the sequence then 8n — 7 = 177

and 8n = 184

n = 184/8 = 23

So “n” **is **a whole number and 177 is in the sequence.

## Quadratic Sequences

The nth term of a quadratic includes an “n²”. To find out if a sequence is quadratic you check the **second **differences. So what is a second difference? We used the first differences between terms when we looked at linear sequences above. **The second difference is the difference between the differences. **This is best understood by using an example. Let’s invent our own quadratic sequence, n² + 1. So looking at the first few terms and the differences and the second differences we have:-

You may get a question where the answer is a multiple or fraction of n². **In these cases the second difference will not be 2, but it will be constant.** For example, find the nth term of this sequence:- 0.25, 1, 2.25, 4, 6.25

Proofs and Proof using algebra to follow.

when doing the nth term with: 0.25, 1, 2.25, 4, 6.25 how did u work it out. i get it but how did it come to 0.25? because in the last one you took the sequence away from the nsquared but now you didn’t?