 In this short video, I just wanted to show the viewers how one could write a repeating decimal as a fraction using a technique of geometric series here. So consider the rational number 1.87 repeated. So the 8-7 after the decimal will be repeated. So this is the number of course 1.87878787, etc. Right? You just repeat this over and over again. And so if you take this number 1.87, well, it can be broken up into your whole number part 1 and your decimal part 8.7 repeated, which like we mentioned earlier 8.7 repeaters means 8 7 8 7 8 7 8 7 8 7 after the decimal place And so every time that this thing repeats itself, we give one period. We're gonna separate that So we're gonna give one the whole number part. We're gonna get point 8 7 point zero zero eight seven point zero zero zero Zero eight seven. Do I say the right number of zeros? There should be four there You're gonna get point zero zero zero zero zero eight seven and then continue this on and so these each of these each of these Right here are terminating decimals that is they stop after a while. We can thus express them You're gonna take the number here. It always could be 87 on top, but then you get powers of 100 Y 100 well 100 is 10 squared and Two here is the length of our period. That's how often this thing repeats itself every two decimal places So we're gonna get one plus 87 over powers of 100 which could be written as a series One plus the sum as where k ranges from one to infinity of 87 over 100 This right here. This is not just any series. This is a geometric series geometric series and Therefore we can use our formula for a geometric series here now when k equals one the first term you're gonna get 87 over 100 That's our first term right here We're using the fact that this thing will look like a over one minus r Where a is the first term in the sequence and r is the constant ratio The constant ratio is gonna be one over 100 that and where 100 here is two squared the length of the period So you get 87 over 100 divided by one minus a hundred one minus a hundred will be 99 over a hundred And then as you simplify this fraction the 100s will actually cancel out and you'll end up with one plus 87 plus 99 over 99 Which if you so choose you can simplify that fraction 87 over 99 becomes 29 over 33 There's common factor of three between 87 and 99 and then writing this mixed number as an improper fraction We get 62 over 33 and so using this geometric series formula one could actually write any any Repeating decimal as a rational number we can find a specific fraction which gives us that repeating decimal