The value of `0. 2^(logsqrt(5)1/4+1/8+1/(16)+)` is `4` b. `log4` c. `log2` d. none of these

Options

A. 4
B. log 4
C. log 2
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \( 0.2^{\left(\log_{\sqrt{5}} \left(\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots\right)\right)} \).

### Step 1: Identify the series
The series \( \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots \) is a geometric series where:
- The first term \( a = \frac{1}{4} \)
- The common ratio \( r = \frac{1}{2} \)

### Step 2: Sum the geometric series
The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values we have:
\[ S = \frac{\frac{1}{4}}{1 - \frac{1}{2}} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \times 2 = \frac{1}{2} \]

### Step 3: Substitute the sum into the logarithm
Now we substitute the sum back into the logarithm:
\[ \log_{\sqrt{5}} \left(\frac{1}{2}\right) \]

### Step 4: Change of base for the logarithm
Using the change of base formula for logarithms:
\[ \log_{\sqrt{5}} \left(\frac{1}{2}\right) = \frac{\log \left(\frac{1}{2}\right)}{\log \left(\sqrt{5}\right)} = \frac{\log \left(\frac{1}{2}\right)}{\frac{1}{2} \log (5)} = \frac{2 \log \left(\frac{1}{2}\right)}{\log (5)} \]

### Step 5: Simplify \( \log \left(\frac{1}{2}\right) \)
We know that:
\[ \log \left(\frac{1}{2}\right) = -\log(2) \]
Thus,
\[ \log_{\sqrt{5}} \left(\frac{1}{2}\right) = \frac{2 \cdot (-\log(2))}{\log(5)} = -\frac{2 \log(2)}{\log(5)} \]

### Step 6: Substitute back into the expression
Now we substitute this back into the original expression:
\[ 0.2^{\left(-\frac{2 \log(2)}{\log(5)}\right)} \]
Since \( 0.2 = \frac{1}{5} \), we can rewrite it as:
\[ \left(\frac{1}{5}\right)^{\left(-\frac{2 \log(2)}{\log(5)}\right)} = 5^{\left(\frac{2 \log(2)}{\log(5)}\right)} \]

### Step 7: Use properties of logarithms
Using the property \( a^{\log_b(c)} = c^{\log_b(a)} \), we can rewrite:
\[ 5^{\left(\frac{2 \log(2)}{\log(5)}\right)} = 2^{\left(2\right)} = 4 \]

### Final Answer
Thus, the value of \( 0.2^{\left(\log_{\sqrt{5}} \left(\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots\right)\right)} \) is:
\[ \boxed{4} \]