Video Transcript
Find the next four terms of the arithmetic sequence represented by the figure.
In this question, we’re given an arithmetic sequence represented by the figure. We need to use this to determine the next four terms of this sequence. To do this, let’s start by recalling what we mean by an arithmetic sequence. And we recall that an arithmetic sequence is a sequence where the difference between any two consecutive terms of the sequence remains constant. Another way of thinking about this is to generate the next term in an arithmetic sequence from the previous term, we just add on a constant value. And there is one more way we can write this. If 𝑎 sub 𝑛 is an arithmetic sequence, then for any positive integer value of 𝑛, 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 must be equal to a constant value of 𝑑. This is called the common difference of the arithmetic sequence.
Before we determine the next four terms of this arithmetic sequence, let’s recall how we graph any sequence. On the 𝑥-axis, we plot the term number, written 𝑛. And on the 𝑦-axis, we plot the corresponding value of that term, 𝑎 sub 𝑛. So the 𝑥-coordinate tells us the term number, and the 𝑦-coordinate tells us the value of that term. So, for example, we can see the point one, four lies on the diagram, which means the first term in our sequence is four. 𝑎 sub one is equal to four. Similarly, since the point with coordinates two, two lies on this diagram, 𝑎 sub two is equal to two. And we can follow the same method to find the values of the third and fourth term of this sequence. It’s the 𝑦-coordinates of the two points. The third term is zero, and the fourth term is negative two.
And now there’s two different ways we can find the next four terms of this sequence. First, we can use what we know about arithmetic sequences. We found the first four terms of this arithmetic sequence to be four, two, zero, negative two. And we can calculate the difference between two consecutive terms of the sequence. For example, the difference between the second and first term of this sequence, two minus four, is equal to negative two. And since this is an arithmetic sequence, we know this will hold true for any two consecutive terms of the sequence. Therefore, we can find the next term of this sequence by subtracting two from the fourth term in our sequence. This gives us the fifth term of our sequence is the fourth term minus two, which is negative two minus two, which we can evaluate is equal to negative four.
We can then use this to find the sixth term in our sequence. This time, we need to subtract two from negative four. And negative four minus two is equal to negative six. We can then follow exactly the same process to find the seventh and eighth term in the sequence. The seventh term in the sequence will be negative six minus two, which is negative eight. And then we subtract two from this value to get the eighth term in the sequence, negative 10. Therefore, we found the next four terms in this arithmetic sequence. They are negative four, negative six, negative eight, and negative 10.
However, this is not the only way we could have answered this question. We can also do this directly from the diagram. To do this directly from the diagram, we can notice all of these points lie on the same straight line. And in fact, for any arithmetic sequence plotted in this manner, all of the points will lie on the same straight line. And in fact, this follows directly from the definition of an arithmetic sequence. To see this, imagine we wanted to find the slope of this straight line. We would want to find the change in 𝑦 divided by the change in 𝑥 for any two points on the line.
One way of doing this would be to use the first two points we’re given. The difference between the 𝑥-coordinates is one, and the difference between the 𝑦-coordinates is negative two. So the slope of the straight line is negative two. And we can find the equation of this straight line by extending it to the vertical axis. We can see that the intercept is at a value of six. This then gives us an equation for the straight line. We’ll write this as 𝑎 sub 𝑛 is equal to negative two 𝑛 plus six. And it’s worth pointing out here we’re only really interested in the positive integer values of 𝑛. This is why we’re writing this as 𝑎 sub 𝑛 for our arithmetic sequence.
So this isn’t really the equation of the entire straight line. It’s just the points on the straight line with positive integer 𝑥-coordinates. But this is all we need to answer our question. What happens if we substitute a positive integer value of 𝑛 into this equation? For example, if we substitute 𝑛 is equal to five into the right-hand side of this equation, we get negative two times five plus six, which, if we evaluate, is equal to negative four, which we know is the fifth term in our sequence. And the reason for this is the slope of our line is negative two, which means for every unit we move across, we will move two units down. Therefore, the points with integer 𝑥-coordinates form an arithmetic sequence.
Therefore, we can also find the next four terms of the arithmetic sequence directly from the diagram. We just mark the next four points of integer 𝑥-coordinates and read off their 𝑦-coordinates. We get negative four, negative six, negative eight, and negative 10. Therefore, we were able to show two different ways of finding the next four terms of the arithmetic sequence given in the figure. Using both methods, we got the next four terms as negative four, negative six, negative eight, and negative 10.
