Divisibility
Task number: 3314
Prove by mathematical induction that \(4 |(6n^2+2n)\).
Solution
For \(n=1\):
\(6{\cdot} 1^2+2{\cdot} 1=8\), so \(4 |(6{\cdot} 1^2+2{\cdot} 1)\)Induction step \(n \to n+1\):
\(6(n+1)^2+2(n+1)=6n^2+12n+6+2n+2=(6n^2+2n)+4(3n+2)\).Because \(4 |(6n^2+2n)\) and also \(4 |4(3n+2)\), we have \(4 |6(n+1)^2+2(n+1)\).
