[Math] Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$

Question

Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$?

I tried graphing but it didn't help. Could anyone answer? Thanks!

Answer 1

\begin{align} f(a) &= \dfrac{a+13}{3a+7} \\ &= \dfrac 13 \dfrac{3a+39}{3a+7} \\ &= \dfrac 13 \left(1 + \dfrac{32}{3a+7} \right) \\ \end{align}

This implies that $f(a)$ is strictly decreasing for positive integers.

Hence the maximum value must be $f(1) = \dfrac 75$