If p+q=66, then which one of the following is correct?

For the following two (02) items: Let p = \sum_{j=1}^{n} \log_{10} 2^j and q = \sum_{j=1}^{n} \log_{10} 5^j.

  1. A. n \lt 7
  2. B. 7 \lt n \lt 9
  3. C. 9 \lt n \lt 12 ✓
  4. D. n \gt 12

Correct Answer: C. 9 \lt n \lt 12

Explanation

We can add the series: p+q = \sum_{j=1}^n (\log_{10} 2^j + \log_{10} 5^j) = \sum_{j=1}^n \log_{10} (10^j) = \sum_{j=1}^n j = \frac{n(n+1)}{2}. Given p+q=66, we have \frac{n(n+1)}{2} = 66 \implies n(n+1) = 132. Thus n=11, which satisfies 9 \lt n \lt 12.

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