Video Transcript
Find the sum of the infinite
geometric sequence seven to the negative one power, negative seven to the negative
three power, seven to the negative five power, and so on.
We’ve been asked to find the sum of
this infinite geometric sequence, in other words, the sum seven to the negative one
power add negative seven to the negative three power add seven to the negative five
power and so on.
Firstly, recall that an infinite
geometric series has the general form 𝑎 one add 𝑎 one 𝑟 add 𝑎 one 𝑟 squared and
so on. We get to the next term by
multiplying the previous term by what we call the common ratio 𝑟. This common ratio is really
important when we’re dealing with infinite geometric series because if 𝑟 is greater
than one, the terms in the sequence get so big that we can’t find an answer for the
sum of the sequence. In fact, we need the absolute value
of 𝑟 to be less than one. And as long as that’s true, then
the sum of an infinite geometric sequence is given by the formula 𝑆 equals 𝑎 one
over one minus 𝑟, where 𝑎 one is the first term of the sequence.
Straightaway, we do know what the
first term of the sequence is, but we don’t know what 𝑟 is. For some sequences like this one,
the common ratio 𝑟 may not be that obvious. But if we take the second term of
the sequence and divide it by the first term of the sequence, we get the common
ratio 𝑟. So for this sequence, we can find
𝑟 by taking the second term, which is negative seven to the negative three power,
and dividing it by the first term, which is seven to the negative one power.
We can then calculate this by
recalling the rule for negative exponents. 𝑎 to the negative 𝑥-power is the
same as one over 𝑎 to the 𝑥-power. So we can rewrite negative seven to
the negative three power as one over negative seven to the third power. And we can rewrite seven to the
negative one power as one over seven.
Note that this is just exactly the
same as saying one over negative seven raised to the third power divided by one over
seven. And we know from the rules of
dividing fractions that this is exactly the same as saying one over negative seven
to the third power times seven over one, which just gives us seven over negative
seven to the third power. But we can actually just calculate
negative seven cubed to be negative 343. We can then divide both the
numerator and the denominator by seven. And we can bring the negative out
to the front of the fraction to give us negative one over 49.
So we’ve calculated 𝑟 to be
negative one over 49. So now we can apply the formula for
finding the sum of an infinite geometric series. That’s 𝑆 equals 𝑎 one over one
minus 𝑟. Remember that 𝑎 one is the first
term of the geometric sequence. So making the substitution, 𝑎 one,
the first term, equals seven to the negative one power and 𝑟 equals negative one
over 49 as we just found. We have that 𝑆 equals seven to the
negative one power over one minus negative one over 49.
Remember that we’ve already said
seven to the negative one power is the same as one over seven. And because we have a double
negative on the denominator, one minus negative one over 49, this is just the same
as one add one over 49. Let’s simplify the denominator a
little bit. One add one over 49 is the same as
49 over 49 add one over 49, which is 50 over 49.
Now, because we have a fraction
over a fraction, remember this is just the same as one over seven divided by 50 over
49. And we know from the rules of
dividing fractions that this is just the same as one over seven times 49 over
50. And we know from the rules of
multiplying fractions by multiplying the numerators and multiplying the
denominators, this gives us 49 over 350. And this in fact cancels down to
seven over 50. And then we have our final
answer. The sum of this infinite geometric
series is seven over 50.
The trickier part of this question
was finding the common ratio 𝑟. In some questions, it may be a
little bit more obvious. But like this question, we might
have to do a little bit more work in order to find 𝑟. But as long as we have 𝑟 and the
first term of the sequence, we can use the formula 𝑆 equals 𝑎 one over one minus
𝑟 to find the sum of the sequence.
