2+5+8+11+⋯+(6n−1)=n(6n+1){\displaystyle 2+5+8+11+\cdots +(6n-1)=n(6n+1)}
L:(6n−1)=6−1=5→2+5=7R:n(6n+1)=6+1=77=7{\displaystyle {\begin{aligned}&L:(6n-1)=6-1=5\rightarrow 2+5=7\\&R:n(6n+1)=6+1=7\\&7=7\\\end{aligned}}}
2+5+8+11+⋯+(6k−1)=k(6k+1){\displaystyle 2+5+8+11+\cdots +(6k-1)=k(6k+1)}
2+5+8+11+⋯+(6k−1)⏟=k(6k+1)+(6k+2)+(6k+5)=(k+1)(6k+7)k(6k+1)+(6k+2)+(6k+5)=(k+1)(6k+7)6k2+k+6k+2+6k+5=6k2+7k+6k+70=0{\displaystyle {\begin{aligned}&\underbrace {2+5+8+11+\cdots +(6k-1)} _{=k(6k+1)}+(6k+2)+(6k+5)=(k+1)(6k+7)\\&k(6k+1)+(6k+2)+(6k+5)=(k+1)(6k+7)\\&6k^{2}+k+6k+2+6k+5=6k^{2}+7k+6k+7\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.