Questions: Express the repeating decimal as a rational number by using a geometric series. (Use symbolic notation and fractions where needed.) 0.636363 ...=

Solution Steps

To express the repeating decimal \(0.636363 \ldots\) as a rational number, we can recognize it as a geometric series. The repeating part is "63", so we can write the decimal as \(0.63 + 0.0063 + 0.000063 + \ldots\). This is a geometric series with the first term \(a = 0.63\) and common ratio \(r = 0.01\). We can use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\) to find the rational number.

The repeating decimal \(0.636363 \ldots\) can be expressed as a geometric series: \[ 0.636363 \ldots = 0.63 + 0.0063 + 0.000063 + \ldots \] Here, the first term \(a = 0.63\) and the common ratio \(r = 0.01\).

The sum \(S\) of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substituting the values of \(a\) and \(r\): \[ S = \frac{0.63}{1 - 0.01} = \frac{0.63}{0.99} \]

Calculating the sum gives: \[ S = 0.6363636363636364 \] To express this as a rational number, we convert \(0.6363636363636364\) to a fraction: \[ S = \frac{7}{11} \]

Final Answer