### Video Transcript

Which of the following is π
geometric sequence? Option (A) π sub π equals three
times π plus three squared, where π is greater than or equal to one. Option (B) π sub π equals π
times three to the power of π minus one, where π is greater than or equal to
two. Option (C) π sub π equals π
times π plus two squared, where π is greater than or equal to one. Option (D) π sub π equals five
times π sub π minus one, where π is greater than or equal to two.

We can begin by remembering that a
geometric sequence is a sequence with a fixed ratio between successive or
consecutive terms. So letβs take each of these
sequences in the given answer options in turn. And it might be useful to find some
of the first terms in each sequence in order to see if there is a fixed ratio
between the terms.

So letβs begin with the sequence
given in option (A), which has this πth term or π sub π as three times π plus
three squared. The first term in this sequence
will be found when π is equal to one. So that means that we can calculate
the term with index one by substituting in a value of π equals one. And so π sub one must be three
times one plus three squared. We can simplify one plus three as
four. And then weβll need to calculate
three times four squared. Well, we know that four squared is
16. And when we multiply that by three,
we get a value of 48. This means that this sequence will
start with a value of 48.

So now, letβs work out the term
which has an index of two. This time, weβll be calculating
three times two plus three squared. And this time, when we simplify two
plus three, we get five, and we know that five squared is 25. So that gives us a second term in
the sequence as 75. We can then work out one other term
in this sequence; that is π sub three. So when we calculate three times
three plus three squared, that will give us a value of 108. So now, what we can do is look at
these first three terms in the sequence and see if we can work out the ratio between
the first term and the second term and the ratio between the second term and the
third term.

If these two ratios are the same,
then that would suggest that it is a geometric sequence. We can work out that if we
multiplied the first term of 48 by the fraction 75 over 48, we would get the second
term of 75. We can simplify this fraction to 25
over 16. Next, to go from the second term to
the third term, we could multiply by 108 over 75. When we simplify this fraction, we
get 36 over 25. We can now easily see that these
two issues are not the same. Because theyβre not the same, then
there is not a fixed ratio between successive terms. And so option (A) is not a
geometric sequence.

We can follow the same process to
see if the sequence given in option (B) is geometric. This time, for the sequence, the
index is given as π is greater than or equal to two. This means that the term will begin
with π sub two. And so when we substitute π is
equal to two into the πth term, we get two times three to the power of two minus
one. When we simplify this, we get a
value of six. For the next term then, thatβs the
term which has an index of three, weβll be calculating three times three to the
power of three minus one. And when we simplify this, we get
an answer of 27.

Next, the term with index four has
a value of 108. We can then calculate the ratios
between these terms. So the first ratio would be 27 over
six. And that simplifies to nine over
two. The next ratio between the terms of
108 and 27 would be simplified to four. If these two ratios were the same
value, that would suggest it is a geometric sequence. However, as they are not, we can
eliminate answer option (B).

For the πth term in option (C), we
could observe that our index here is π is greater than or equal to one. So letβs calculate the terms π sub
one, π sub two, and π sub three. When we substitute the values of π
equals one, two, and three into the πth term formula, we get values of nine, 32,
and 75. We can once again look at the
ratios between these three terms. We can work these out as 32 over
nine and 75 over 32. Neither of these fractions
simplifies any further. And we can see that theyβre not
equal. So option (C) is not a geometric
sequence.

Finally, letβs have a look at the
sequence which is given in option (D). This sequence is slightly different
in that itβs an example of a recursive sequence. Because we have this πth term as
π sub π, then the term π sub π minus one is the term before π sub π. In fact, if we wanted to describe
this πth term in words, we could say that we find any term π sub π by multiplying
the term before by five. If we wanted to find the first
terms in this sequence, we may have a bit of a problem. Thatβs because even though we know
that our index π is greater than or equal to two, when we go to find π sub two, we
know that it must be five times π sub one.

But weβre not actually told what π
sub one is. Usually, when weβre given the πth
term of a sequence written in a recursive manner, then we also need to be told the
value of the first term in the sequence or the term which has an index of one. But letβs see if it actually
matters for this question.

We know that our sequence will have
the terms π sub two, π sub three, and π sub four. π sub two will be five π sub one,
π sub three will be equal to five π sub two, and π sub four will be five π sub
three. That means that the ratio between
any successive term is five. And this means that we do have a
sequence which has a fixed ratio between successive terms. Therefore, we can give the answer
that the sequence π sub π equals five π sub π minus one where π is greater than
or equal to two is a geometric sequence.