Evaluate the infinite geometric series $0.79 + 0.079 + 0.0079 + 0.00079 + 0.000079 + \dotsb$. Express your answer as a fraction with integer numerator and denominator.
You must use S=a/1-r
so you plug the number in S=.79/1-.1
which is S=.79/.9 so we have 79/90 as our answer
btw S=sum a= first number in sequence r=common ratio
hope this makes sense
To evaluate the infinite geometric series $0.79 + 0.079 + 0.0079 + 0.00079 + 0.000079 + \dotsb$, we can use the formula for the sum of an infinite geometric series:
\[S = \frac{a}{1 - r}\]
where:
- $S$ represents the sum of the series,
- $a$ represents the first term of the series, and
- $r$ represents the common ratio between consecutive terms.
In this case, the first term, $a$, is $0.79$, and the common ratio, $r$, can be found by dividing any term by its previous term. For example:
\[r = \frac{0.079}{0.79} = \frac{0.0079}{0.079} = \frac{0.00079}{0.0079} = \frac{0.000079}{0.00079} = \dotsb\]
Since the common ratio between consecutive terms is constant, we can safely calculate the value of $r$ using any of these divisions. In this case, $r = 0.1$.
Plugging the values of $a$ and $r$ into the formula, we get:
\[S = \frac{0.79}{1 - 0.1}\]
Simplifying the denominator, we have:
\[S = \frac{0.79}{0.9}\]
To express this sum as a fraction with an integer numerator and denominator, we multiply both the numerator and denominator by $100$ to remove the decimal:
\[S = \frac{0.79 \times 100}{0.9 \times 100} = \frac{79}{90}\]
Therefore, the value of the infinite geometric series is $\frac{79}{90}$.