What is the next term of the quadratic sequence shown below? 18, 17, 13, six.
There are several types of sequence that we need to be able to recognize for our GCSE. Those are linear sequences, quadratic sequences, geometric sequences, and Fibonacci sequences.
Linear sequences have a constant first difference. For example, the sequence one, three, five, seven is going up by two each time. That’s its first difference.
Quadratic sequences have a constant second difference. We’ll see what that looks like in a moment. But a good example of a quadratic sequence is the sequence of square numbers.
Geometric sequences have a common ratio. For example, the sequence one, two, four, eight has a common ratio of two. Each number is being multiplied by two.
And then, the Fibonacci sequence is a sequence where the next term is found by adding the previous two terms. One plus one is two. One plus two is three. Two plus three is five. So the next term in this sequence will be three plus five which is eight.
Since we’re told that this is a quadratic sequence, we know we’re going to need to find the second difference. To do that, we’ll begin by finding the first difference.
What do we do to get from 18 to 17? We subtract one. So the first difference here is negative one. To get from 17 to 13, we subtract four. And to get from 13 to six, we subtract seven.
We can see that the first difference changes each time. That’s just confirm that this can’t be a linear sequence. Now, let’s find the second difference between the terms.
To get from negative one to negative four, we subtract three. And to get from negative four to negative seven, we also subtract three. And now, we have our constant second difference. It’s negative three.
This means the next second difference also has to be negative three. This will allow us to find the next value for the first difference. And in turn, we’ll be able to find the next term in the sequence. To find the next first difference, we need to subtract three from negative seven.
A common difference is to think of this gives an answer of negative four. Let’s see what’s happening on the number line. We start at negative seven. We’re going to subtract three. That means we move down the number line three spaces: one, two, three. We end up at negative 10.
So negative seven minus three is negative 10. And that’s the value of our next first difference. To find the next term in the quadratic sequence, we’re going to need to then subtract 10 from six.
Remember when we subtract, it’s another way of saying what’s the difference between these numbers. Well, we know the difference between 10 and six is four. In this example, we’re subtracting the larger number from the smaller number. So that means the difference between six and 10 must be negative four.
And we found the next number in our sequence. The fifth term in this quadratic sequence is negative four.