In this section we’re going to be looking at quadratic sequences. We shall explore how to find the nth term of a quadratic sequence.
Working with quadratic sequence
Here is an example of a sequence;
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.
Here is another quadratic sequence example;
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;
- When the second difference is 2 the sequence starts with n2
- When the second difference is 4, the sequence starts with 2n2
- When the second difference is 8, the sequence starts with 4n2
And so on…