Answered: Use the geometric series to find the…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter7: Exponents And Exponential Functions

Section7.9: Geometric Sequences As Exponential Functions

Problem 26PPS

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Question

Use the geometric series to find the rational numbers with 7-adic expansions 2.421¯ 7 and 3.¯67.

Expert Solution

Step 1

Consider the number $5.421 \bar{7}$

To use geometric series to determine the ration numbers for the expansions, write the decimal number in the form as:

$5.421 + 0.0007 + 0.00007 + 0.000007 + . . . . . . . . . .$

Now write the fractional forms for all the terms.

$\frac{5421}{1000} + \frac{7}{10000} + \frac{7}{100000} + \frac{7}{1000000} + . . . . . . . . . . . .$

Here consider the geometric series from the second term.

$\frac{7}{10^{4}} + \frac{7}{10^{5}} + \frac{7}{10^{6}} + . . . . . . . . . . . .$

For this geometric series, the common ratio is $\frac{1}{10}$ with initial term as $\frac{7}{10^{4}}$ .

The geometric sum of the series is thus;

$\frac{(\frac{7}{10^{4}})}{1 - (\frac{1}{10})} = \frac{7}{9000}$

Now add this to the first term $\frac{5421}{1000}$

$\begin{array}{rcl} \frac{5421}{1000} + \frac{7}{9000} & = & \frac{48789}{9000} + \frac{7}{9000} \\ = & \frac{48796}{9000} \\ = & \frac{12199}{2250} \end{array}$

Therefore, the rational form of $5.421 \bar{7}$ is $\frac{12199}{2250}$ .

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