To solve this problem, we need to find the sum of the given geometric sequence:

$S = \frac{1}{5} + \frac{4}{5} + \frac{4^2}{5} + \frac{4^3}{5} + \cdots + \frac{4^{10}}{5}.$

Step 1: General Form of the Geometric Sequence

The sequence can be written as: $a_n = \frac{4^{n-1}}{5}, \quad n = 1, 2, \dots, 11.$

Step 2: Formula for the Sum of a Finite Geometric Sequence

The formula for the sum of the first $n$ terms of a geometric sequence is: $S_n = a \frac{1 - r^n}{1 - r},$ where:

• $a$ is the first term,
• $r$ is the common ratio,
• $n$ is the number of terms.

Step 3: Identify Parameters

• The first term $a = \frac{1}{5}$,
• The common ratio $r = \frac{4}{5}$,
• The number of terms $n = 11$.

Step 4: Substitute Values

Substituting these into the formula: $S_{11} = \frac{1}{5} \cdot \frac{1 - \left(\frac{4}{5}\right)^{11}}{1 - \frac{4}{5}}.$

The denominator simplifies: $1 - \frac{4}{5} = \frac{1}{5}.$

The formula becomes: $S_{11} = \frac{1}{5} \cdot \frac{1 - \left(\frac{4}{5}\right)^{11}}{\frac{1}{5}}.$

The $\frac{1}{5}$ in the numerator and denominator cancels out: $S_{11} = 1 - \left(\frac{4}{5}\right)^{11}.$

Step 5: Compute Values

1. Calculate $\left(\frac{4}{5}\right)^{11}$: $\left(\frac{4}{5}\right)^{11} = 0.0264241152 \quad (\text{approximately}).$

2. Subtract from 1: $S_{11} = 1 - 0.0264241152 = 0.9735758848.$

3. Round to two decimal places: $S_{11} \approx 0.97.$

Final Answer:

The sum of the geometric sequence is: $\boxed{0.97}$

Would you like me to verify this computation or explain further?

Here are some related questions to expand your understanding:

1. What happens to the sum if the number of terms $n$ is increased to infinity?
2. How do you determine the convergence of a geometric sequence?
3. Can a geometric series ever have a negative sum? Why or why not?
4. How is the sum formula derived for a finite geometric sequence?
5. Can you rewrite this sequence in sigma notation?

Tip: Always double-check your calculations with a calculator or graphing utility when dealing with powers and fractions.