Video Transcript
Is the sequence 𝑎 sub 𝑛 is equal to negative seven 𝑛 minus 62 increasing or decreasing?
In this question, we’re given an expression for the 𝑛th term of a sequence. And we need to determine whether this sequence is increasing or decreasing. To do this, let’s start by recalling what we mean by an increasing and a decreasing sequence. First, we recall we say that a sequence is increasing if its terms are getting 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. The next term always needs to be bigger than the previous term. Similarly, we recall we say that a sequence is decreasing if the terms are getting smaller and smaller. This means for any positive integer value of 𝑛, 𝑎 sub 𝑛 plus one must be smaller than 𝑎 𝑛. The next term is always smaller than the previous term.
Let’s take a look at some of the terms of this sequence by substituting values of 𝑛 into our expression. We can find the first term of this sequence by substituting 𝑛 is equal to one into this expression. The first term is negative seven times one minus 62, which we can calculate. It’s equal to negative 69. We can do the same to find the second term of this sequence. We substitute 𝑛 is equal to two to get negative seven multiplied by two minus 62, which we can calculate is equal to negative 76. And we can immediately notice negative 76 is smaller than negative 69. Therefore, the second term is smaller than the first term. So our sequence can’t be increasing.
We may want to just conclude that our sequence is decreasing. However, for due diligence, we should show this is true for any positive integer value of 𝑛. And there’s a few different ways of doing this. One way is to consider the inequality we have for decreasing sequences. We can rearrange this inequality to have 𝑎 𝑛 plus one minus 𝑎 𝑛 is less than zero. In other words, we can also show that a sequence is decreasing if we can show the difference between a term of the sequence and its previous term is always negative. So let’s find an expression for this difference.
We’ll start by finding an expression for the term in position 𝑛 plus one. It’s given by negative seven times 𝑛 plus one minus 62. We can distribute over our parentheses and simplify to get negative seven 𝑛 minus 69. And we’re already given an expression for the 𝑛th term of our sequence. 𝑎 sub 𝑛 is negative seven 𝑛 minus 62. We want to find an expression for 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛. This will then give us an expression for the difference between any two consecutive terms of the sequence.
We can find this by substituting in the expressions we found above. We get negative seven 𝑛 minus 69 minus negative seven 𝑛 minus 62. We can distribute the negative over our parentheses. We get negative seven 𝑛 minus 69 plus seven 𝑛 plus 62, which we can evaluate is equal to negative seven. And we can see that this value is negative. In other words, the difference between any term in our sequence and the previous term is negative. Our sequence must be decreasing. Therefore, we were able to show the sequence 𝑎 sub 𝑛 is equal to negative seven 𝑛 minus 62 is decreasing.
