Question

If \[ \sum_{k=1}^{n}\log_{10}(5^k)=66\log_{10}(5), \] then the value of \(n\) is equal to:

• A. 8
• B. 9
• C. 10
• D. 11
• E. 12

Hint:

To solve \(n(n+1) = 132\) quickly, note that \(\sqrt{132} \approx 11.5\). The integer value of \(n\) will be the floor of this square root.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The problem involves a summation of logarithmic terms. We can use the property of logarithms \(\log(a^b) = b \log(a)\) and the sum of the first \(n\) natural numbers to simplify the equation.

Step 2: Key Formula or Approach:
1. Logarithmic property: \(\log_{10}(5^k) = k \log_{10}(5)\).
2. Sum of first \(n\) natural numbers: \(\sum_{k=1}^n k = \frac{n(n+1)}{2}\).

Step 3: Detailed Explanation:
The given equation is:
\[ \sum_{k=1}^n \log_{10} (5^k) = 66 \log_{10}(5) \]
Using the property \(\log(5^k) = k \log(5)\):
\[ \sum_{k=1}^n k \log_{10}(5) = 66 \log_{10}(5) \]
Since \(\log_{10}(5)\) is a constant, we can factor it out of the summation:
\[ \log_{10}(5) \sum_{k=1}^n k = 66 \log_{10}(5) \]
Dividing both sides by \(\log_{10}(5)\) (which is non-zero):
\[ \sum_{k=1}^n k = 66 \]
Using the sum formula:
\[ \frac{n(n+1)}{2} = 66 \]
\[ n(n+1) = 132 \]
We need two consecutive integers whose product is 132.
\[ 11 \times 12 = 132 \]
Thus, \(n = 11\).

Step 4: Final Answer:
The value of \(n\) is 11.