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# Infinite geometric series word problem: repeating decimal

AP.CALC:
LIM‑7 (EU)
,
LIM‑7.A (LO)
,
LIM‑7.A.3 (EK)
,
LIM‑7.A.4 (EK)

## Video transcript

let's say we have the repeating decimal zero point four zero zero eight where the digits four zero zero eight keep on repeating so if we were to write it out it would look something like this zero point four zero zero eight four zero zero eight four zero zero eight and 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 four zero zero eight so for example I will make this four zero zero eight my first term so this could be viewed as zero and these this four zero zero eight represents zero point four zero zero eight then I could make this for zero zero eight my next term or my next term will represent this for zero zero eight and that means its zero and this for zero zero eight is the same thing as zero point zero zero zero zero four zero zero eight and then this next for zero zero eight well that represents zero point and we have eight zeros one two three four five six seven eight four zero zero eight four zero zero eight 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 this this is there's a pattern here we're essentially throwing four zeros before the decimal every time and and we could just keep on going like that forever so this is an infinite sum it's an infinite series the next question is 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 zero point four zero zero eight to this one right over here where we add four zeros before the four zero zero eight what are we multiplying well we move the decimal four spots to the left so we're multiplying by to the negative 4th or you could view it as we're multiplying by 0.001 absolute or one more 0 0 1 10 to the negative 1 2 3 4 to go from here to here well same thing move the decimal 4 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 4th power so we can rewrite we can rewrite all of this business as zero point 4 0 0 8 times our common ratio for this first term times our common ratio of 10 to the negative fourth to the 0th power so that's that gives us that right over there plus zero point 4 0 0 8 times 10 to the negative fourth to the first power and that gives us that value right over there plus zero point 4 0 0 8 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 to infinity of well what's our first term going to be it's going to be zero point four zero or zero point four zero zero eight times times our common ratio which we could write out as either 10 to the negative fourth or 0.0001 I'll try it as 10 to the negative fourth 10 to the negative fourth to the K power to the K power so the next interesting question we've now this is clearly can be represented as a geometric series is well what is this sum you know you might say well that's just going to be four zero zero eight repeating over and over but I want to express it as a fraction and so I want you to pause this video use what you already know about try 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 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 zero to infinity of a R 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 zero point four zero zero eight and it's going to be that over 1 1 minus our common ratio - and I'll write it like this zero point zero zero zero one one ten-thousandth so what's this going to be well this is going to be the same thing as zero point four zero zero eight if you if you take one minus one ten-thousandth or you could use this ten thousand ten thousandth minus one ten thousandth you're going to have you're going to have 9999 over or 9,999 10,000 ten thousands once again you could view let me write this out just so this doesn't look confusing one is the same thing as ten thousand over ten thousand and you're subtracting one over ten thousand and so you're going to get 9999 over ten thousand and so this is going to be the same thing this is going to be the same thing as zero point four zero zero eight times ten thousand so times ten thousand over nine thousand nine hundred and ninety nine well what's this top number times ten thousand well that's just going to give us four thousand and eight four thousand and eight over nine thousand nine hundred ninety nine nine thousand nine hundred 9 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 what if we can if this is already a fraction so we've already kind of achieved it but if we want to get a little bit simpler see if we add the digits up here 4 plus 8 is 12 so in 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 4008 let's see it goes into 4 one time subtract you get a 10 3 times 3 is 9 subtract yet another 10 goes into 3 times 3 times 3 is 9 subtract bring down an 8 3 goes into 18 exactly 6 times so our numerator is 1336 this is no longer divisible by 3 the sum of the digits is not divisible by 3 is not a multiple of 3 and if you divide this bottom number by 3 you get 3333 and I think we have simplified it I think we have simplified it about as well as we can although I might 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