Question 100
Let $n \ge 1$ be an integer and consider the sum
$$
x = \sum_{k \ge 0} \binom{n}{2k} 2^{n-2k} 3^k = \binom{n}{0} 2^n + \binom{n}{2} 2^{n-2} \cdot 3 + \binom{n}{4} 2^{n-4} \cdot 3^2 + \dots
$$
Show that $2x - 1$, $2x$, $2x + 1$ form the sides of a triangle whose area and inradius are also integers.
$$
x = \sum_{k \ge 0} \binom{n}{2k} 2^{n-2k} 3^k = \binom{n}{0} 2^n + \binom{n}{2} 2^{n-2} \cdot 3 + \binom{n}{4} 2^{n-4} \cdot 3^2 + \dots
$$
Show that $2x - 1$, $2x$, $2x + 1$ form the sides of a triangle whose area and inradius are also integers.