Video Transcript
Is the sequence two, negative three, negative eight, negative 13, and so on increasing or decreasing?
In this question, we’re given a sequence and we need to determine whether this sequence is increasing or decreasing. We can start by recalling what we mean by increasing and decreasing sequences. An increasing sequence is a sequence where the terms get larger and larger. In other words, for any positive integer value of 𝑛, 𝑎 sub 𝑛 plus one must be bigger than 𝑎 sub 𝑛. Similarly, a decreasing sequence is one where the terms get smaller and smaller. For any positive integer value of 𝑛, 𝑎 sub 𝑛 plus one must be smaller than 𝑎 sub 𝑛.
So, to determine if this sequence is increasing or decreasing, let’s see what happens to the terms of this sequence. First, we can see that two is bigger than negative three. So the second term in this sequence is smaller than the first term in this sequence. Another way of seeing this is to find the difference between the two terms. The second term minus the first term is negative five. We’ve decreased the first term by negative five. We can do the same with the second and third term of the sequence. Negative eight is smaller than negative three. In fact, it’s five smaller than negative three. And once again, the exact same is true for the third and fourth term of the sequence. The fourth term is smaller than the third term, and it’s five smaller. We decrease negative eight by five to get negative 13.
Therefore, since the terms of the sequence are getting smaller and smaller, we can conclude that this is a decreasing sequence. However, there is something worth pointing out here. We can see the difference between any two consecutive terms remains constant. The difference between consecutive terms is negative five. And sequences where the difference between consecutive terms has a name. They’re called arithmetic sequences. And the constant difference is called the common difference. And in an arithmetic sequence, we can find the next term in the sequence from the previous term by adding on the common difference. For example, we can find the second term in the sequence by adding the common difference of negative five to the first term. Two plus negative five is negative three.
We can then notice something interesting. When our common difference is negative, we’re decreasing the values of our sequence. And when this value is positive, we’re increasing the values of this sequence. And this allows us to show a very useful property. If we have an arithmetic sequence with common difference 𝑑, then to find the next term in the sequence, we add the value of 𝑑 to the previous term. So, when 𝑑 is positive, our sequence will be increasing, because we increase the previous term by 𝑑. And similarly, when 𝑑 is negative, we decrease the value of the previous term. So, when 𝑑 is negative, the sequence is decreasing.
Therefore, we were able to show the sequence two, negative three, negative eight, negative 13, and so on is a decreasing sequence.
