To find the value of \( q - p \) given \( p + q = 15 \), we must first understand the definitions provided:

We have:

• \( p = \sum_{j=1}^n \log_{10} 2^j \)
• \( q = \sum_{j=1}^n \log_{10} 5^j \)

The expression for \( p \) can be simplified using the properties of logarithms:

• \( p = \sum_{j=1}^n j \log_{10} 2 \)

Similarly, for \( q \):

• \( q = \sum_{j=1}^n j \log_{10} 5 \)

Adding these two, we have:

• \( p + q = \sum_{j=1}^n j(\log_{10} 2 + \log_{10} 5) \)

Using the property \(\log_{10} 2 + \log_{10} 5 = \log_{10} 10 = 1\), this becomes:

• \( p + q = \sum_{j=1}^n j \times 1 = \sum_{j=1}^n j \)

The sum of the first \( n \) natural numbers is given by:

• \( \frac{n(n+1)}{2} = 15 \)

This implies \( p + q = 15 \), which matches the given condition.

Now we need to find \( q - p \):

• \( q - p = \sum_{j=1}^n j(\log_{10} 5 - \log_{10} 2) \)

Using the property \( \log_{10} a - \log_{10} b = \log_{10} \left(\frac{a}{b}\right) \), we have:

• \( q - p = \sum_{j=1}^n j \log_{10} \left(\frac{5}{2}\right) \)

Thus, \( q - p \) becomes:

• \( \frac{n(n+1)}{2} \log_{10} \left(\frac{5}{2}\right) \)

Given \( p + q = 15 \), we know it simplifies finally to \(15 \log_{10} 2.5\), hence the correct answer is:

Answer: \( 15 \log_{10} 2.5 \)