By finding the sum of an infinite geometric sequence, express 0.13 recurring plus 0.732 recurring as a common fraction.
Let’s begin by recalling the formula for finding the sum of an infinite geometric sequence. It’s 𝑎 sub one over one minus 𝑟. 𝑎 sub one represents the first term in the sequence, and 𝑟 represents the common ratio. And in order for this sum to be convergent, the absolute value of 𝑟 must be strictly less than one. We’ve been given two recurring decimals and asked to find their sum as a common fraction by considering the sum of an infinite geometric sequence. We could try to express each of these recurring decimals individually as infinite geometric sequences. But as the question just says an — i.e., one — infinite geometric sequence, let’s first consider their sum in decimal form.
First, 0.13 recurring is the decimal 0.13131313 continuing, and this is because the overline is above both the one and the three. So both digits recur. For the second decimal, the overline is only above the three and the two, so this is the decimal 0.732323232 continuing. If we now add in columns, we get the decimal 0.86363636 continuing. What we have then is another recurring decimal, 0.863 recurring. And we’ll consider just how we can find the single recurring decimal as the sum of an infinite geometric sequence.
Let’s separate out the digits of this decimal a little. We can then write it as 0.8 plus 0.063 plus 0.00063 plus 0.0000063 and so on. All of the terms other than the 0.8 are 100 times smaller than the previous term. And so, they form a geometric sequence with a common ratio 𝑟 of one over 100, or 0.01. The first term in this sequence is 0.063. And so, we can substitute the values for 𝑎 sub one and 𝑟, which does have an absolute value strictly less than one into the infinite sum formula. This gives 0.063 over one minus 0.01. That’s 0.063 over 0.99. And then we can multiply both the numerator and denominator here by 1000 so that they’re both integers. That gives 63 over 990. And we can then divide both the numerator and denominator of this fraction by nine to give the simplified fraction seven over 110.
We’ve worked out then that the sum of this infinite geometric sequence is seven over 110. But in order to find this entire decimal as a fraction, we need to include the 0.8 at the beginning. As a fraction, this is equal to eight over 10. So we have the sum of eight over 10 and seven over 110. Multiplying both the numerator and denominator of the fraction eight over 10 by 11, we have 88 over 110. And we’re adding seven over 110 to this value; that’s 95 over 110. And then we can divide both the numerator and denominator by five to give 19 over 22.
This is our answer to the problem then. By finding the sum of an infinite geometric sequence, we’ve expressed the decimal sum 0.13 recurring plus 0.732 recurring as the fraction 19 over 22.