The value of `[(0.16)^(log_(2.5)(1/3+1/3^2+1/3^3+….+oo))]^(1/2)` is a) 1 b) 2 c) 3 d) -1

Options

A. `1`
B. `2`
C. `3`
D. `-1`

Explanation

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

### Step 1: Calculate the sum of the infinite series
The series \(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots\) 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 is given by the formula:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \]

**Hint:** Remember that the sum of an infinite geometric series converges when the absolute value of the common ratio is less than 1.

### Step 2: Substitute the sum into the logarithm
Now we substitute this sum into the logarithmic expression:
\[ \log_{2.5}\left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots\right) = \log_{2.5}\left(\frac{1}{2}\right) \]

**Hint:** Ensure you understand how to evaluate logarithms and how to substitute values into them.

### Step 3: Rewrite \(0.16\) in a simpler form
Next, we rewrite \(0.16\):
\[ 0.16 = \frac{16}{100} = \frac{4}{25} \]

**Hint:** Converting decimals to fractions can simplify calculations.

### Step 4: Substitute into the expression
Now we can substitute this back into our original expression:
\[ \left[\left(\frac{4}{25}\right)^{\log_{2.5}\left(\frac{1}{2}\right)}\right]^{\frac{1}{2}} = \left(\frac{4}{25}\right)^{\frac{1}{2} \log_{2.5}\left(\frac{1}{2}\right)} \]

**Hint:** Pay attention to the properties of exponents when simplifying expressions.

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

**Hint:** Familiarize yourself with logarithmic identities to manipulate expressions effectively.

### Step 6: Evaluate the logarithm
Now we evaluate \(\log_{2.5}\left(\frac{4}{25}\right)\):
\[ \frac{4}{25} = \frac{2^2}{5^2} \implies \log_{2.5}\left(\frac{4}{25}\right) = \log_{2.5}(2^2) - \log_{2.5}(5^2) = 2\log_{2.5}(2) - 2\log_{2.5}(5) \]

**Hint:** Break down logarithms into simpler components to evaluate them.

### Step 7: Final simplification
Putting everything together:
\[ \left(\frac{1}{2}\right)^{\log_{2.5}\left(\frac{4}{25}\right)} = \left(\frac{1}{2}\right)^{2\log_{2.5}(2) - 2\log_{2.5}(5)} \]
This simplifies to:
\[ \left(\frac{1}{2}\right)^{\log_{2.5}(2^2) - \log_{2.5}(5^2)} = \left(\frac{1}{2}\right)^{\log_{2.5}\left(\frac{4}{25}\right)} \]

### Step 8: Evaluate the entire expression
Finally, we can evaluate:
\[ \left(\frac{1}{2}\right)^{-2} = 2^2 = 4 \]

Thus, the value of the original expression is:
\[ \boxed{2} \]