Convert 8.690909 . . . into a single fraction using infinite geometric series.
8.690909....
=8.6 + .09 + .0009 + .000009+...
= 86/10 + .09/(1-.01)
= 86/10 + .09/.99
= 86/10 + 9/99
= 86/10 + 1/11
= 478/55
To convert the repeating decimal 8.690909... into a single fraction using an infinite geometric series, we can follow these steps:
Step 1: Identify the repeating pattern
Looking at the decimal, we can observe that the repeating pattern is 90. Therefore, we need to focus on this pattern to form our series.
Step 2: Express the repeating pattern as a fraction
Since the repeating pattern is 90, we can express it as 90/100, which simplifies to 9/10.
Step 3: Express the series using the formula
The formula for summing an infinite geometric series is: S = a / (1 - r), where S represents the sum, a is the first term, and r is the common ratio.
In our case, a = 8 (since 8.690909... begins with 8), and r = 1/10 (since the repeating pattern is 9/10).
Step 4: Calculate the sum
Plugging in the values into the formula, we find:
S = 8 / (1 - 1/10)
S = 8 / (9/10)
S = 80 / 9
Therefore, the fraction representation of the decimal 8.690909... is 80/9.