Question

Let \[ \alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty \] and \[ \beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty. \] 
Then the value of \[ (0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)} \] is equal to: ________

• A. 4
• B. 5
• C. 8
• D. 25

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
First, calculate the infinite geometric progression sums \(\alpha\) and \(\beta\). Then, substitute these into the logarithmic expression using exponential and logarithm base conversion properties.

Step 2: Key Formula or Approach:
1. \(S_\infty = \frac{a}{1 - r}\).
2. \(a^{\log_b c} = c^{\log_b a}\).
3. \(\log_{b^k} x = \frac{1}{k} \log_b x\).

Step 3: Detailed Explanation:
\(\alpha = \frac{1/4}{1 - 1/2} = \frac{1}{2}\).
\(\beta = \frac{1/3}{1 - 1/3} = \frac{1}{2}\).
Expression: \((0.2)^{\log_{\sqrt{5}}(1/2)} + (0.04)^{\log_{5}(1/2)}\).
Term 1: \((5^{-1})^{\frac{\log_5(1/2)}{\log_5(5^{1/2})}} = (5^{-1})^{2 \log_5(1/2)} = 5^{-2 \log_5(1/2)} = 5^{\log_5 4} = 4\).
Term 2: \((5^{-2})^{\log_5(1/2)} = 5^{-2 \log_5(1/2)} = 5^{\log_5 4} = 4\).
Sum \(= 4 + 4 = 8\).

Step 4: Final Answer:
The total value is 8.