Proving an Inequality

[math7839]

I would like to prove that n^2 + 63 is greater than or equal to 16n for all positive integers n. I have tried to do this by setting n^2 + 63 = 16n + x and also by using properties of inequalities, but I get stuck both ways. Please help!

 

[soroban]

Hello, math7839!

The inequality is not true . . .

\(\displaystyle \text{Prove: }\;n^2 + 63 \: \ge \:16n\,\text{ for all }n \in N.\)

\(\displaystyle \begin{array}{ccccc}\text{We have:} & n^2 - 16n + 63 & \begin{array}{c}> \\ [-2mm] <\end{array} & 0 \\ \text{Add 1:} & n^2 - 16n + 64 & \begin{array}{c}> \\ [-2mm] <\end{array} & 1 \\ \text{Factor:} & (n-8)^2 & \begin{array}{c}> \\ [-2mm] <\end{array} & 1 \end{array}\)


\(\displaystyle \text{The left side is greater than or equal to 1 }except\text{ when }n = 8.\)

 

[math7839]

Thanks so much! Can't believe I missed that.