Exercise 58 Page 697

We want to write the repeating decimal 0.7 as a fraction. To do so, we will start by writing this number as a sum of decimals and as a sum of fractions.

Consider the sum of fractions above. Note that we can think of it as a geometric series that has a first term of a_1= 710. To find its common ratio r, we can divide any term of the sequence by its previous term. For simplicity, we will divide a_2 by a_1.

Each term of the sequence can be obtained by multiplying the previous term by the common ratio 110.

Let's substitute r= 110 and a_1= 710 in the formula for the sum of an infinite geometric series.

We found that the series formed by the sum of the terms is 79. With this information we can express the given number as a single fraction. 0.7=7/9