### Video Transcript

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

In this question, we need to
calculate the πth term or general term of this given sequence. The value of π indicates the
position that the term has in the sequence. So letβs take our index π 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 πth term is. However, as an alternative method,
letβs imagine that we donβt 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 letβs 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 π. 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 π, 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 πth term of this sequence has a value of three π
on the numerator and a value of π on the denominator. But of course, sometimes sequences
like this can take a while to figure out. So letβs see how we could use the
answer options to help us work this out instead.

Letβs consider the general term
which is given in answer option (A) three to the power of π minus one. Since we were given the first three
terms in this unknown sequence, then letβs compare it with the first three terms
which would be generated by the sequence with πth term three to the power of π
minus one. We can find the first term by
substituting π 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 letβs check answer option
(B).

Although this looks very similar to
the πth term in answer option (A), the difference here is that one is subtracted
from the value π 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 π equals two, which gives us three squared over two, which is nine
over two. For the third term, we substitute
π 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
πth term in answer option (C) must be the correct one. And finally, for completeness, when
we generate the sequence given by the πth 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 πth term of this sequence is three to the power of π
over π.