Quadratic Sequences

The differences between these terms in the sequences are not equal. This must mean that the sequence is not linear as it does not increase in equal steps as shown below;

We can see that the difference of this sequence changes at each step as it increases. Suppose we found the second difference;

The second difference of the sequence is constant. It does not change as the first difference. If the second difference is constant the sequence is referred to as a quadratic sequence, and therefore contains a n² term.

Here is the sequence again and the terms labelled;

In the sequence above we can see that the first term is a 4, second term is a 7, third term is 19, and the fifth term is 28. Let’s find n² below;

If you look at the sequence you will realise that the sequence is always 3 more than n² term as shown below;

This must mean that the rule to find any nth term in this sequence is;

We can prove this;

The above shows that our rule to find the nth term in the above sequence is valid.

First we find the first difference as we did above;

Then we find the second difference;

We can see that the second difference has a constant increment of 4. This must mean that it’s a quadratic sequence and therefore as an n²

Now it’s find the n² next and compare;

Here we can see that when we compare the n² sequence with the original sequence the outcome is not constant, we have to change n² to make sure that it does. Let’s try 2n²;

Now we have found a link by using 2n². We can see that the sequence is always 5 more than the 2n² sequence. Therefore the formula to find any nth term in this sequence is;

In the last example there was a twist. And you must familiarise yourself with this. In the first sequence we saw that the second difference was a 2 therefore we use n². In the last example the second difference was 4 we found that 2n² gave us a link to use with the sequence. You must use this trial and improvement method when this occurs to make sure that the link is always constant. If you continue to practice you will realise that;

And so on… Therefore you always halve the second difference to get the number of n²s for example if the second difference of a sequence is +4 (which would be constant) you would divide 4 by 2 to get 2 therefore this becomes 2n²

We take the same steps as we did above. We have to find the first and second difference.

Look at the second difference we can see that just 1 n² is involved since…

We continue as we did before by finding the difference between n² and the sequence terms

The residue is not constant. We therefore have to find the nth term of the residue. Given that;

We simply add the residue nth term to n². So given the sequence;

…the nth term of the sequence is;

First find the first difference and then the second difference of the sequence;

Since the second difference is constant that must mean at the sequence is a quadratic sequence. Therefore the sequence involves n² We divide the second difference by two to determine the number of n² involved;

Therefore the nth term starts with 5n²…

We must now work out how the the sequence is from n²

Notice that the residue is not constant here. We therefore have to find the nth term of the residue.

Therefore;

You can prove it for your self by finding all the terms in the original sequence.