\text{Assume that }\sqrt{4n-2} = p/q,\ \text{where } p,q\in \mathbb{Z}, q\neq 0\ \text{and}\ \operatorname{gcd}(p,q) = 1 \newline \begin{align*} &\implies 4n-2 = p^2/q^2 \\&\implies 2(2n-1)q^2 = p^2 \\&\implies p^2\ \text{is even} \\&\implies p\ \text{is even} \\&\implies p = 2a\ \text{for...