Chapter 9.4, Problem 31E

Solution

To calculate: A fraction of the rational number $7.14141414\cdots$ .

The resultant fraction is $\frac{707}{99}$ .

Given information:

The given rational number is $7.14141414\cdots$ .

Formula used: The geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a\cdot r^{k-1}}$ , converges if and only if $\left|r\right|<1$ .If it does converge, the sum: $S=\frac{a_1}{1-r}$ .

Calculation:

Consider the given rational number $7.14141414\cdots$ ,

Rewrite $7.14141414\cdots$ as,

  $\begin{array}{c}7.14141414\cdots =7+\left(0.14141414\cdots \right)\\ =7+\left(0.14+0.0014+0.000014+\cdots \right)\end{array}$

Now $0.14+0.0014+0.000014+\cdots$ has become a geometric series. The terms are:

  $a_1=0.14,a_2=0.0014,a_3=0.000014$

The common ratio is:

  $r=\frac{0.0014}{0.14},r=0.01$

Use the formula for sum $\frac{a_1}{1-r}$ , and substitute $0.14$ for $a_1$ and 0.01 for r ,

  $\begin{array}{c}\frac{a_1}{1-r}=\frac{0.14}{1-0.01}\\ =\frac{0.14}{0.99}\\ =\frac{14}{99}\end{array}$

Now find the value of $7+\left(0.14+0.0014+0.000014+\cdots \right)$ ,

  $\begin{array}{c}7+\left(0.14+0.0014+0.000014+\cdots \right)=7+\frac{14}{99}\\ =\frac{7\cdot 99+14}{99}\\ =\frac{693+14}{99}\\ =\frac{707}{99}\end{array}$

Thus, the repeating decimals $7.14141414\cdots$ can be written in fraction as $\frac{707}{99}$ .