Find, in terms of 𝑛, the general term of the sequence 18, 72, 162, 288, and so on. Option (A) 18𝑛, option (B) 18𝑛 squared, option (C) 18𝑛 cubed, option (D) 19𝑛 minus one, or option (E) 17𝑛 plus one.
In this question, we need to work out the general term of this sequence given in terms of 𝑛. The value of 𝑛 will represent the index or position of each term. So let’s take the value of 𝑛 to be integers greater than or equal to one, which means that the first term will have an index one. The second term would then have an index of two, the third with an index of three, and so on.
We can use the answer options to help us work out what this general term could be. So let’s start with the possibility in answer option (A) that the general term is 18𝑛. So let’s work out what the sequence 18𝑛 would actually look like. For the first term, we would multiply 𝑛, which is one, by 18. And that would give us a value of 18. So far, this is matching our sequence. So let’s see what happens when the index is two. Well, this time, we’re multiplying 18 by two, and that would give us a value of 36. However, the second term in the given sequence is 72. So this means that the sequence that we were given is not 18𝑛. And so we can therefore eliminate answer option (A).
So let’s look at answer option (B). This time, let’s work out what a sequence with a general term of 18𝑛 squared would produce. The first term in the sequence will be calculated as 18 multiplied by one squared. Well, we know that one squared is one, and 18 times one will give us 18. The second term is calculated as 18 multiplied by two squared. And two squared is four. 18 times four is 72. When we work out the third and fourth terms as 18 times three squared and 18 times four squared, respectively, we get the values of 162 and 288. And so we’ve worked out that the sequence that is produced by the general term 18𝑛 squared will be identical to those in the given sequence. Therefore, the answer is that given in option (B). The general term of the sequence 18, 72, 162, 288, and so on is 18𝑛 squared.
Although the other answer options in (C), (D), and (E) would produce the same first term, none of these general terms would produce the entire sequence that we were given.
There is also another way in which we could approach this problem. When we’re given an unknown sequence, we might perhaps see if there’s a common ratio or a common difference between terms. If there is a fixed ratio between terms, that would indicate a geometric sequence. In order to identify a ratio between the first terms, we would think, what do we multiply 18 by to get 72? Or alternatively we divide 72 by 18. And this would give us an answer of four. If there is a common ratio, then when we multiply 72 by four, we would get 162. However, we would not get a value of 162. In fact, the ratio between 72 and 162 is nine over four. Therefore, there is not a fixed ratio between the terms in this sequence. And so this is not a geometric sequence.
We might then establish if this is an arithmetic sequence, in which case there would be a common difference between the terms in the sequence. However, when we calculate the difference between terms, we can see that the difference changes every time. And so this is not an arithmetic sequence either.
However, there is one further type of sequence that we might wish to check. And that is to establish if this is a quadratic sequence. In a quadratic sequence, the difference of the differences will be constant. When we work this out, we can see that the differences between the differences is 36. Therefore, we can say that this given sequence will be a quadratic sequence. And so when written in terms of 𝑛, then this sequence will relate to 𝑛 squared.
However, if we simply had the sequence 𝑛 squared, then the first term would be one squared, which is one. The second term would be two squared, which is four. The third term would be nine, and the fourth term would be 16. And we know that we don’t have simply 𝑛 squared because we have the sequence which starts with 18. And so since the first term in the given sequence is 18, then the most sensible thing to check would be if 18𝑛 squared would generate the given sequence. And we’ve already established that it does.
And so either by using the answer options or by working out what type of sequence we have, we can say that the general term of this sequence is 18𝑛 squared.