Exercise 59 Page 697

We want to write the repeating decimal 5.126 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 from the 2^(nd) term on, we can think of it as a geometric series that has a first term of a_1= 1261000. 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 11000.

Let's substitute r= 11000 and a_1= 1261000 in the formula for the sum of an infinite geometric series.

We found that the series formed by the sum of the terms, starting from the 2^(nd), is 14111. With this information we can express the given number as a sum of two fractions. 5.126=5/1+14/111 Finally, we will add the fractions to obtain the value of 5.126 expressed as a single fraction.