This activity is about finding a rule to generate a quadratic sequence.

**To find the nth term rule for an arithmetic sequence:**

Let's take the sequence **3, 5, 7, 9, ..........**

The first difference is **+2** as we can see that the next term is 2 more than the previous term.

**This means the sequence goes up in the 2 times table!**

We call this **2n**

However, the first term in the 2 times table is 2 and we want 3, so we have to **add 1 to all the terms.**

**The rule to find any term in that sequence is 2n + 1**

Let's look at a quadratic sequence now!

What we mean by a** quadratic sequence** is that it has** n ^{2 } **in the rule.

Unlike the arithmetic sequence above, it **does not increase** by the same amount each time!

Let's look at an example!

Find the nth term rule for the quadratic sequence:

**2, 5, 10, 17, 26, ............**

**Method**

We look at the n^{2 }sequence.

That is 1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2},..........etc

Which is, 1, 4, 9, 16, 25, ..........

We can see that the sequence is very similar.

What is the difference?

We can see that if we subtract the n^{2} sequence, the sequence we have is 1 more each time.

Therefore, the rule is **n ^{2} + 1**

It looks a bit complicated, but when you have done a couple, it gets easy very quickly!

Let's give it a go!