2. Write the repeating decimal $$ 0 . \overline{09}=0.0909090909 \ldots $$ first as a geometric series and then as a fraction (a ratio of two integers).

Question

UpStudy AI · Accepted Answer

To express the repeating decimal $$ 0.\overline{09} $$ as a geometric series, we can break it down as follows:

1. **Geometric Series Representation:**
   The repeating decimal $$ 0.090909...$$ can be expressed as:
   $$
   0.090909... = 0.09 + 0.0009 + 0.000009 + \ldots
   $$
   This series can be recognized as a geometric series where:
   - The first term $$ a = 0.09 $$
   - The common ratio $$ r = 0.01 $$

The sum $$ S $$ of an infinite geometric series can be calculated using the formula:
   $$
   S = \frac{a}{1 - r}
   $$

2. **Calculating the Sum:**
   Now, substituting the values of $$ a $$ and $$ r $$:
   $$
   S = \frac{0.09}{1 - 0.01}
   $$

Let's calculate this expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.09}{1-0.01}$$
- step1: Subtract the numbers:
  $$\frac{0.09}{0.99}$$
- step2: Convert the expressions:
  $$\frac{\frac{9}{100}}{\frac{99}{100}}$$
- step3: Multiply by the reciprocal:
  $$\frac{9}{100}\times \frac{100}{99}$$
- step4: Reduce the numbers:
  $$1\times \frac{1}{11}$$
- step5: Multiply:
  $$\frac{1}{11}$$
The sum of the geometric series is:

$$
S = \frac{0.09}{1 - 0.01} = \frac{0.09}{0.99} = \frac{1}{11}
$$

Thus, the repeating decimal $$ 0.\overline{09} $$ can be expressed as the fraction $$ \frac{1}{11} $$.

In summary:
- As a geometric series: $$ 0.09 + 0.0009 + 0.000009 + \ldots $$
- As a fraction: $$ \frac{1}{11} $$
The repeating decimal $$ 0.\overline{09} $$ can be expressed as the fraction $$ \frac{1}{11} $$.

Write the repeating decimal first as a geometric series and then
as a fraction (a ratio of two integers).

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Pre Algebra Questions

Write the repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).