Step 1: Section Formula for X-coordinate:
Let point R be \( \left(\frac{8}{5}, \frac{18}{5}\right) \). A is \( (2, p/2) \), B is \( (p, -2p) \).
\[ x_R = \frac{mx_2 + nx_1}{m+n} \implies \frac{8}{5} = \frac{mp + 2n}{m+n} \]
\[ 8m + 8n = 5mp + 10n \]
\[ m(8-5p) = 2n \implies \frac{m}{n} = \frac{2}{8-5p} \]
Step 2: Analyze Integer Constraint:
The y-coordinate check yields non-integer/imaginary values for the given point coordinates directly (likely typo in question point values). However, using the constraint "p is an integer" and matching options usually implies a consistent integer solution exists for the expression.
Let's check the expression value with the derived ratio.
Expression: \( E = \frac{3m+n}{3n} = \frac{m}{n} + \frac{1}{3} \).
Substitute \( \frac{m}{n} \):
\[ E = \frac{2}{8-5p} + \frac{1}{3} \]
If the answer is Option (A) \( p \):
\[ \frac{2}{8-5p} + \frac{1}{3} = p \implies 6 + (8-5p) = 3p(8-5p) \]
\[ 14 - 5p = 24p - 15p^2 \]
\[ 15p^2 - 29p + 14 = 0 \]
\[ (15p-14)(p-1) = 0 \]
Since \( p \) is an integer, \( p=1 \).
This gives a valid integer solution.
Checking other options yields non-integer values for \( p \). Thus, the relationship holds for \( p=1 \).
Step 3: Final Answer:
The value of the expression corresponds to \( p \).