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# Infinite geometric series word problem: repeating decimal

Video transcript

Let's say we have the
repeating decimal 0.4008, where the digits 4008
keep on repeating. So if we were to
write it out, it would look something like this. 0.400840084008, and it
keeps on going forever. What I want you to do right
now is pause the video and think about whether
you can represent this repeating decimal
as an infinite sum, as an infinite series. And then think about
whether that infinite series is a geometric series. So I'm assuming you've
given a go at it. So let's think about it. So for each term of
my infinite series, I'm going to represent one
of these repeating patterns of 4008. So, for example, I will make
this 4008 my first term. So this could be viewed as 0. And this 4008 represents 0.4008. Then I could make this
4008 my next term, or my next term will
represent this 4008. And this 4008 is the
same thing as 0.00004008. And then this next
4008, well, that represents 0 point-- and
we have eight zeroes-- 1, 2, 3, 4, 5, 6,
7, 8, 4008, 4008. And then we would just keep
on going like that forever. So we're just going to keep
on going like that forever. So hopefully there's
a pattern here. We're essentially throwing
four zeroes before the decimal every time. And we can just keep on
going like that forever. So this is an infinite sum. It's an infinite series. The next question is, is
this a geometric series? Well, in order for it to
be a geometric series, to go from one term
to the next, you must be multiplying
by the same value, by the same common ratio. So what are we multiplying when
we go from 0.4008 to this one right over here, where we add
four zeros before the 4008? What are we multiplying? Well, we moved the decimal
four spots to the left. So we're multiplying by
10 to the negative fourth. Or you could view it
as we're multiplying by 0.0001, 10 to the
negative 1, 2, 3, 4. To go from here to
here, well, same thing. Move the decimal four
places to the left. So once again, we're
multiplying by 0.0001. And so it looks
pretty clear that we have a common ratio of 10 to
the negative fourth power. So we can rewrite
all of this business as 0.4008 times our common
ratio for this first term times our common ratio of
10 to the negative fourth to the 0-th power--
so that gives us that right over there--
plus 0.4008 times 10 to the negative fourth
to the first power. And that gives us that
value right over there. Plus 0.4008 times 10 to the
negative 4 to the second power. And we keep on going. And so in this form, it
looks a little bit clearer, like a geometric series, an
infinite geometric series. And if we wanted to write
that out with sigma notation, we could write this
as the sum from k equals 0 to infinity
of, well, what's our first term going to be? It's going to be 0.4008
times our common ratio, which we could write out as either
10 to the negative fourth or 0.0001. I'll just write it as 10
to the negative fourth. 10 to the negative fourth to the
k-th power, to the k-th power. So the next
interesting question-- this clearly can be represented
as a geometric series-- is, well, what is the sum? You might say, well,
that's just going to be 4008 repeating
over and over. But I want to express
it as a fraction. And so I want you
to pause the video. Use what would you
already know about finding the sum of an infinite
geometric series to try to express this thing
right over here as a fraction. So I'm assuming
you've had a go at it. So let's think about it. We've already
seen, we've already derived in previous
videos, that the sum of an infinite
geometric series-- let me do this in
a neutral color. If I have a series like this,
k equals 0 to infinity of ar to the k power, that this sum
is going to be equal to a over 1 minus r. We've derived this actually
in several other videos. So in this case, this
is going to be-- well, our a here is 0.4008. And it's going to be that over 1
minus our common ratio, minus-- and I'll write it like this--
0.0001, 1 ten thousandth. So what's this going to be? Well, this is going to be
the same thing as 0.4008. If you take 1 minus
1 ten thousandths, or you could do this as
10,000 ten thousandths minus 1 ten thousandth, you're going
to have 9,999 ten thousandths. Once again, you could
view-- let me write this out just so this doesn't
look confusing. 1 is the same thing
as 10,000/10,000. And you're subtracting 1/10,000. And so you're going
to get 9,999/10,000. And so this is going
to be the same thing as 0.4008 times 10,000. So times 10,000 over 9,999. Well, what's this top
number times 10,000? Well that's just going
to give us 4,008. 4,008 over 9,999. And we've just expressed
that repeating decimal as a fraction. So we have succeeded. And you might say, well, maybe
we can simplify this thing. And so let's see. This is already a
fraction, so we've already kind of achieved it. But if we want to get
a little bit simpler. Let's see, if we add the
digits up here, 4 plus 8 is 12 and 1 plus 2 is 3. So this up here
is divisible by 3. And this down here is
clearly divisible by 3. So let's divide
both of them by 3. So 3 goes into
4,008-- let's see. It goes into 4 one time,
subtract, you get a 10. 3 times 3 is 9. Subtract, you get another 10. Goes into 3 times. 3 times 3 is 9. Subtract. Bring down an 8. 3 into 18 exactly six times. So our numerator is 1,336. This is no longer
divisible by 3. The sum of the digits
is not divisible by 3, it's not a multiple of 3. And if you divide this bottom
number by 3, you get 3,333. And I think we
have simplified it. I think we can simplified
it about as well as we can. Well, we could check more. Let me know if I didn't. But either way, we
have now written this. This was pretty neat. We saw that a
repeating decimal can be represented not just
as an infinite series, but as an infinite
geometric series. And then we were able to use
the formula that we derived for the sum of an
infinite geometric series to actually express
it as a fraction.