Question 15
Let $f(n)$ and $g(n)$ be functions satisfying $$
\begin{aligned} &f(n) = \\ &\quad \begin{cases} \sqrt{n} & \substack{\text{if } \sqrt{n} \text{ is} \\ \text{an integer}} \\ 1 + f(n+1) & \text{otherwise} \end{cases} \end{aligned}
$$ and $$
\begin{aligned} &g(n) = \\ &\quad \begin{cases} \sqrt{n} & \substack{\text{if } \sqrt{n} \text{ is} \\ \text{an integer}} \\ 2 + g(n+2) & \text{otherwise} \end{cases} \end{aligned}
$$ for positive integers $n.$ Find the least positive integer $n$ such that $\frac{f(n)}{g(n)} = \frac{4}{7}.$
\begin{aligned} &f(n) = \\ &\quad \begin{cases} \sqrt{n} & \substack{\text{if } \sqrt{n} \text{ is} \\ \text{an integer}} \\ 1 + f(n+1) & \text{otherwise} \end{cases} \end{aligned}
$$ and $$
\begin{aligned} &g(n) = \\ &\quad \begin{cases} \sqrt{n} & \substack{\text{if } \sqrt{n} \text{ is} \\ \text{an integer}} \\ 2 + g(n+2) & \text{otherwise} \end{cases} \end{aligned}
$$ for positive integers $n.$ Find the least positive integer $n$ such that $\frac{f(n)}{g(n)} = \frac{4}{7}.$