The value of `(0*16)^(log_(2*5)((1)/(3)+(1)/(3^(2))+(1)/(3^(3))+....oo))` is

Options

A. `4`
B. `3`
C. `2`
D. none

Answer

To solve the expression \( (0.16)^{\log_{2.5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots\right)} \), we will follow these steps:

### Step 1: Calculate the sum of the infinite series
The series given is:
\[ \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots \]
This is a geometric series where the first term \( a = \frac{1}{3} \) and the common ratio \( r = \frac{1}{3} \).

The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

### Step 2: Substitute the sum into the logarithm
Now we substitute the sum back into the logarithm:
\[ (0.16)^{\log_{2.5}\left(\frac{1}{2}\right)} \]

### Step 3: Rewrite \( 0.16 \)
Next, we can express \( 0.16 \) in terms of powers of \( 2 \):
\[ 0.16 = \frac{16}{100} = \frac{16}{10^2} = \frac{2^4}{(2 \cdot 5)^2} = \frac{2^4}{2^2 \cdot 5^2} = \frac{2^2}{5^2} = \frac{4}{25} \]

### Step 4: Rewrite the expression
Now we can rewrite our expression:
\[ \left(\frac{4}{25}\right)^{\log_{2.5}\left(\frac{1}{2}\right)} \]

### Step 5: Use the properties of logarithms
Using the property of logarithms \( a^{\log_b(c)} = c^{\log_b(a)} \), we can rewrite the expression:
\[ \left(\frac{4}{25}\right)^{\log_{2.5}\left(\frac{1}{2}\right)} = \left(\frac{1}{2}\right)^{\log_{2.5}\left(\frac{4}{25}\right)} \]

### Step 6: Calculate \( \log_{2.5}\left(\frac{4}{25}\right) \)
We can express \( \frac{4}{25} \) as:
\[ \frac{4}{25} = \frac{2^2}{5^2} \]
Thus,
\[ \log_{2.5}\left(\frac{4}{25}\right) = \log_{2.5}(2^2) - \log_{2.5}(5^2) = 2\log_{2.5}(2) - 2\log_{2.5}(5) \]

### Step 7: Substitute back and simplify
Now substituting back:
\[ \left(\frac{1}{2}\right)^{2\log_{2.5}(2) - 2\log_{2.5}(5)} = \left(\frac{1}{2}\right)^{2(\log_{2.5}(2) - \log_{2.5}(5))} \]
This can be simplified further, but we can see that the expression will evaluate to a numerical value.

### Final Result
The final value of the expression is:
\[ \left(\frac{1}{2}\right)^{0} = 1 \]