<s>[INST] A simple quadratic inequality For positive integers $n\ge c\ge 5$, why does
$$c+2(n-c)+\frac{(n-c)^2}{4}\le\frac{(n-1)^2}{4}+1\text{ ?}$$
 [/INST] If you expand the two sides, you will get
\begin{equation}
\frac{c^2}{4}-\frac{cn}{2}-c+\frac{n^2}{4}+2n\leq\frac{n^2}{4}-\frac{n}{2}+\frac{5}{4}
\end{equation}
\begin{equation}
\frac{c^2}{4}-\frac{5}{4}-c\leq n\frac{c-5}{2}
\end{equation}
If c=5 than the inequality is satisfied, assuming $5\leq c$ we have
\begin{equation}
\frac{c-5}{2} -\frac{2c}{c-5}\leq n
\end{equation}
I think you can use induction to prove this inequality.
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