To find the sum of the infinite geometric series [tex]\( 25 - 5 + 1 - \frac{1}{5} + \frac{1}{25} - \ldots \)[/tex], follow these steps:
### Step 1: Identify the first term ([tex]\( a \)[/tex]) and the common ratio ([tex]\( r \)[/tex])
1. The first term [tex]\( a \)[/tex] of the series is given as [tex]\( 25 \)[/tex].
2. To find the common ratio [tex]\( r \)[/tex], look at the ratio of the second term to the first term:
[tex]\[
r = \frac{-5}{25} = -\frac{1}{5}
\][/tex]
### Step 2: Verify that the series converges
The series converges if the absolute value of the common ratio [tex]\( |\mathbf{r}| \)[/tex] is less than 1. In this case:
[tex]\[
\left| -\frac{1}{5} \right| = \frac{1}{5} < 1
\][/tex]
Since [tex]\( |\mathbf{r}| < 1 \)[/tex], the series converges.
### Step 3: Use the formula for the sum of an infinite geometric series
The formula for the sum [tex]\( S \)[/tex] of an infinite geometric series is:
[tex]\[
S = \frac{a}{1 - r}
\][/tex]
Substitute [tex]\( a = 25 \)[/tex] and [tex]\( r = -\frac{1}{5} \)[/tex] into the formula:
[tex]\[
S = \frac{25}{1 - \left( -\frac{1}{5} \right)}
\][/tex]
### Step 4: Simplify the expression inside the denominator
Simplify the denominator first:
[tex]\[
1 - \left( -\frac{1}{5} \right) = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}
\][/tex]
Then, substitute back into the sum formula:
[tex]\[
S = \frac{25}{\frac{6}{5}} = 25 \times \frac{5}{6} = \frac{125}{6} \approx 20.833333333333336
\][/tex]
### Step 5: Round the answer to the nearest hundredth
Rounded to the nearest hundredth, the sum [tex]\( S \)[/tex] is:
[tex]\[
S \approx 20.83
\][/tex]
Thus, the sum of the infinite geometric series [tex]\( 25 - 5 + 1 - \frac{1}{5} + \frac{1}{25} - \ldots \)[/tex], rounded to the nearest hundredth, is:
[tex]\[
\boxed{20.83}
\][/tex]
