Video Transcript
Find the common ratio of an
infinite geometric sequence given that the sum is 52 and the first term is 14.
For this question, we have been
asked to find the common ratio of an infinite geometric sequence. The first thing we should note is
that sequences and series are closely interlinked. We have been told the sum of our
sequence is 52. At this point, we remember that the
sum of an infinite sequence is what defines an infinite series. In essence, we’ll need to use the
tools for an infinite geometric series for this question. The first thing we can do is to
recall the general form for an infinite geometric series. This type of series can be
characterized by a first term 𝑎 and a common ratio 𝑟 for which successive terms
are found by multiplying the previous term by the common ratio.
Now, a general rule that we can use
for geometric series is if the absolute value of the common ratio 𝑟 is less than
one, then the series is convergent. If this is the case, then the value
of the sum is equal to the first term 𝑎 divided by one minus 𝑟, the common
ratio. If instead the absolute value of 𝑟
is greater than or equal to one, then the series is divergent. And of course, for a divergence
series we cannot assign this sum a value. Now returning back to our question,
the first thing we can note is that the question has indeed assigned the sum a
finite value. This implies the series is
convergent, and we can ignore the case of a divergent series. It also means that our sum can be
expressed as 𝑎 divided by one minus 𝑟. And of course, this is equal to
52.
The other piece of information that
we’ve been given by our question is that the first term is 14. This is great since we can use the
given value for 𝑎 to find the unknown value for 𝑟 using our equation. We first substitute 𝑎 equals
14. We now solve for 𝑟, which is the
common ratio that the question is asking for. We can multiply both sides by one
minus 𝑟, multiply both the terms in parentheses by 52. We then subtract 52 from both sides
of our equation. And for our final step, we can
divide both sides by negative 52. Doing so, we find the value of 𝑟,
the common ratio, is 19 divided by 26. With this step, we have answered
our question. We used the given information and
our knowledge of geometric series and how these relate to geometric sequences to
find that the common ratio is 19 over 26.
