By finding the sum of an infinite geometric sequence, express 3.72 recurring as a common fraction.
The key to this question is the part that’s circled. We have the digits seven and two with a line above them, which means the seven and two are recurring. Let’s begin by recalling what we mean by a recurring decimal. There are a couple of common ways of writing a recurring decimal: firstly, as in this case, with a line above the numbers that are repeating or recurring. Alternatively, we write a dot above each of the digits that are recurring. In either case, 0.72 recurring is equal to 0.727272 and so on. There are several ways of rewriting a recurring decimal as a fraction.
In this question, we are asked to do so by finding the sum of an infinite geometric sequence. We begin by splitting 0.72 recurring as follows. It is equal to 0.72 plus 0.0072 plus 0.000072 and so on. The right-hand side of our equation is an example of a geometric series, where the first term is 𝑎, the second term 𝑎𝑟, and the third term 𝑎𝑟 squared. The terms of the series could be rewritten as a geometric sequence with first term 𝑎 equal to 0.72. We can calculate the value of the common ratio 𝑟 by dividing any term by the preceding one, for example, dividing 0.0072 by 0.72. This is equal to one hundredth, or 0.01. We now have a geometric sequence with first term 0.72 and common ratio 0.01, which is equivalent to the recurring decimal 0.72 recurring.
Since the absolute value of the common ratio is less than one, we can find the sum of this infinite geometric sequence. The sum to infinity is equal to the first term 𝑎 divided by one minus 𝑟. Substituting in our values of 𝑎 and 𝑟, we have 0.72 divided by one minus 0.01. This is equal to 0.72 over 0.99, which in turn simplifies to 72 over 99. Both the numerator and denominator here have a common factor of nine. And as 72 divided by nine is eight and 99 divided by nine is 11, our fraction simplifies to eight over 11, or eight elevenths.
The sum to infinity of our geometric sequence is eight elevenths. This means that the fraction eight elevenths is equal to the decimal 0.72 recurring. 3.72 recurring is therefore equal to three and eight elevenths. And as three is equal to thirty-three elevenths, we have thirty-three elevenths plus eight elevenths. This is equal to 41 over 11, or forty-one elevenths.
3.72 recurring, written as a common fraction, is forty-one elevenths.