Question Video: Finding the General Term of a Given Sequence Mathematics • Second Year of Secondary School

Video Transcript

The 𝑛th term of the sequence three, nine over two, nine, and so on is what. Option (A) three to the power of 𝑛 minus one, option (B) three to the power of 𝑛 minus one, option (C) three to the power of 𝑛 over 𝑛, or option (D) three to the power of 𝑛 minus 𝑛.

In this question, we need to calculate the 𝑛th term or general term of this given sequence. The value of 𝑛 indicates the position that the term has in the sequence. So let’s take our index 𝑛 to be integers which are greater than or equal to one, which means that the first term of the sequence would have an index of one. The second term has an index of two and so on. Here, we are given four different answer options which we could use to establish what the 𝑛th term is. However, as an alternative method, let’s imagine that we don’t have any answer options and we can treat this like a puzzle that we need to solve.

The first thing we notice is that we have integer values and fractional values. We might also notice that the term which has an index of two is a fraction that has a denominator of two. So what if in fact every term in this sequence at least starts off as a fraction? This would mean that the first term will be a fraction which simplifies to three, and the third term would be a fraction which simplifies to nine. So let’s think separately about the numerators and denominators of each term in this sequence.

The second term we definitely know is a fraction nine over two. Perhaps the denominators of this sequence is simply a sequence of a general term 𝑛. If it was, then we would have the sequence one, two, three on the denominators. For the numerators, we might look at the values and realize that there are a few threes or multiples of threes. We might even wonder if there are powers of threes. If we consider the numerators to have a sequence of general term three to the power of 𝑛, then the first term would be three to the power of one, which would give us a value of three. The second term would therefore be three squared, which would give us nine. And the third term would be three to the power of three, which is 27.

If we then put together the numerators and denominators to form a single value, then this would give us the sequence three over one, nine over two, 27 over three. And since three over one simplifies to three and 27 over three simplifies to nine, then this is identical to the given sequence. And therefore, we could give the answer that the general term or 𝑛th term of this sequence has a value of three 𝑛 on the numerator and a value of 𝑛 on the denominator. But of course, sometimes sequences like this can take a while to figure out. So let’s see how we could use the answer options to help us work this out instead.

Let’s consider the general term which is given in answer option (A) three to the power of 𝑛 minus one. Since we were given the first three terms in this unknown sequence, then let’s compare it with the first three terms which would be generated by the sequence with 𝑛th term three to the power of 𝑛 minus one. We can find the first term by substituting 𝑛 equals one into this formula. And three to the power of one is three subtract one would give us two. We could continue if we wished, but we can already see that two does not match the first term of three in this given sequence. And so we can eliminate answer option (A). So let’s check answer option (B).

Although this looks very similar to the 𝑛th term in answer option (A), the difference here is that one is subtracted from the value 𝑛 in the exponent. And when we calculate three to the power of zero like any value to the power of zero, it gives us an answer of one. And once again, this first term does not match the first term of three in the given sequence, and so we can eliminate answer option (B). So when we check answer option (C), the first term will be generated by three to the power of one over one. Three to the power of one is three and divided by one will give us three.

The second term is found by substituting 𝑛 equals two, which gives us three squared over two, which is nine over two. For the third term, we substitute 𝑛 equals three. Three cubed is 27 and divided by three gives us a value of nine. Since these three terms of three, nine over two, and nine do indeed match the given sequence, then we can say that the 𝑛th term in answer option (C) must be the correct one. And finally, for completeness, when we generate the sequence given by the 𝑛th term in answer option (D), we would get the values two, seven, and 24. And since these do not match the given terms, then we can eliminate answer option (D). Therefore, either method allows us to give the answer that the 𝑛th term of this sequence is three to the power of 𝑛 over 𝑛.