Video Transcript
In a geometric sequence, the
ratio between any two successive terms is a fixed ratio 𝑟. Consider the sequence one-half,
one-quarter, one-eighth, one sixteenth, and so on. Is this sequence geometric? Consider the sequence one-half,
one-quarter, one-eighth, one sixteenth, and so on. What is the value of 𝑟? Consider the sequence one-half,
one-quarter, one-eighth, one sixteenth, and so on. What is the general term of
this sequence?
In this question, we’re given a
reminder of what a geometric sequence is. It’s one which has a fixed
ratio or common ratio between any two successive terms. In each of the three parts of
this question, we’re considering the same sequence. And in the first part of this
question, we’re asked if this given sequence is geometric. So let’s write down this
sequence. If it is geometric, then there
will be a common ratio 𝑟 between any two consecutive or successive terms. So let’s see if we can find a
ratio between the first two terms, one-half and one-quarter. To find the ratio, we take the
second term of one-quarter and divide it by one-half.
When we’re dividing fractions,
we write the first fraction and we multiply it by the reciprocal of the second
fraction. We can take out a common factor
of two from the numerator and denominator. And then multiplying the
numerators gives us one, and multiplying the denominators gives us two. That means that the ratio
between the first and second term is one-half. Next, we will find the ratio
between the second and third term of one-quarter and one-eighth, so we’ll
calculate one-eighth divided by one-quarter. Multiplying by the reciprocal
of the fraction one-quarter, we calculate one-eighth multiplied by four over
one. And so, once again, we get the
ratio of one-half.
It looks like we probably do
have a common ratio, but it’s always worth checking all the terms to make sure
that there is a common ratio on all of them. And when we calculate one
sixteenth divided by one-eighth, we also get the ratio of one-half. Therefore, we have established
that this sequence does have a fixed or common ratio, and so it must be
geometric. And we can say yes as the
answer for the first part of this question. In the second part of this
question, we’re asked to find the value of 𝑟 for the same sequence. Remember that 𝑟 is the fixed
ratio and it’s nice and simple. We have just worked it out to
be one-half, which is the second part of this question answered.
In the final part of this
question, we’re asked to find the general term of this sequence. And we can remember that the
general term is another way of asking for the 𝑛th term. If we started with the first
term written as 𝑎 sub one equal to one-half, then the second term would be 𝑎
sub two and it would be one-quarter. The third and fourth terms can
be written as 𝑎 sub three and 𝑎 sub four. So when we’re finding the
general term, we’re really looking for the rule that would allow us to work out
the 𝑛th term or 𝑎 sub 𝑛.
We can remember that there is a
general formula to allow us to work out the 𝑛th term of any geometric
sequence. 𝑎 sub 𝑛 is equal to 𝑎
multiplied by 𝑟 to the power of 𝑛 minus one, where 𝑎 is the first term and 𝑟
is the common ratio. The values that we need to plug
into the formula will be 𝑎 equals one-half, as that was the first term, and 𝑟
we worked out as one-half. Therefore, the 𝑛th term of
this sequence can be given as a half multiplied by one-half to the power of 𝑛
minus one. When we give our answer, it’s
important that we indicate the values of 𝑛. When we find the 𝑛th term or
general term, we started with 𝑛 equals one. So the answer for the third
part of the question is that the general term of the sequence is one-half
multiplied by one-half to the power of 𝑛 minus one for values of 𝑛 greater
than or equal to one.
We can, of course, further
simplify this general term using one of the rules of exponents. If we consider that the first
value of one-half is equivalent to one-half to the power of one, and so adding
the exponents one and 𝑛 minus one will give us simply the exponent of 𝑛. If we then plugged in the
values of 𝑛 equals one, two, three, or four into either of these formulas, we’d
get the first four terms of the sequence that we were given, and so verifying
that we have the correct answer for the general term.
