For the given sequence, what is the missing term? And then we have a sequence whose first term is negative 60, whose third term is negative 2,160, the fourth term is 12,960, and the fifth term is negative 77,760.
Now we’re looking to find the second term of this sequence, so we need to begin by working out what kind of sequence we’ve actually been given. To do so, we’ll look at the relationship between the third, fourth, and fifth terms in this sequence. Now there are a number of different types of sequences that we could be looking at. For instance, we could have a quadratic sequence or even a Fibonacci sequence. But it’s always sensible to begin by looking for an arithmetic or geometric sequence to begin with.
Remember, an arithmetic sequence is one in which the difference between each consecutive term is constant, and a geometric sequence has a common ratio between consecutive terms. And, of course, the constant difference 𝑑 will simply be the difference between any two consecutive terms in the sequence, so 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛, where 𝑛 is no smaller than one. Similarly, the common ratio is found by dividing any term by the term that precedes it. So this time that’s 𝑎 sub 𝑛 plus one divided by 𝑎 sub 𝑛.
So, let’s begin by asking ourselves, is there a common difference? And to do so, we’ll subtract the third term from the fourth and then the fourth term from the fifth. If these two values are equal, then we can assume that we are given an arithmetic sequence. The difference between the third and fourth term is 15,120. Then the difference between the fourth and fifth term is negative 90,720. These are not equal to one another, so there is not a common difference and we can deduce that this is not an arithmetic sequence.
So next, we’ll ask ourselves, is there a common ratio? Let’s divide the fourth term by the third and the fifth term by the fourth. The ratio between the third and fourth terms is negative six. Similarly, the ratio between the fourth and fifth terms is also negative six. Since there is a common ratio, we can deduce that this must be a geometric sequence. Now, of course we are making an assumption, and that is that the pattern continues no matter which direction we head.
So, now that we have decided that we have a geometric sequence, how do we find the second term in this sequence? Well, we can use the definition or we can use a formula. The definition of a geometric sequence, or part of it, is that each term is found by multiplying the previous term by this common ratio. So, to get to the second term, we multiply the first term by the common ratio. To get to the third term, we multiply the second term by the same common ratio, and so on. Alternatively, and this comes from the very definition, the 𝑛th term of a geometric sequence is found by multiplying the first term 𝑎 by the common ratio to the power of 𝑛 minus one.
Either way, we find the second term by multiplying the first term by negative six. So, 𝑎 sub two is negative 60 times negative six, which is 360. And whilst we made an assumption earlier on that the pattern continues, we can double-check by multiplying 360 by negative six and checking that we get the value of the third term. 𝑎 sub three is 360 times negative six, which is indeed negative 2,160. So, once we’ve calculated that we have a geometric sequence with a common ratio of negative six, we can find the missing term to be 360.