2 + 5 + 8 + 11 + ⋯ + ( 6 n − 1 ) = n ( 6 n + 1 ) {\displaystyle 2+5+8+11+\cdots +(6n-1)=n(6n+1)}
L : ( 6 n − 1 ) = 6 − 1 = 5 → 2 + 5 = 7 R : n ( 6 n + 1 ) = 6 + 1 = 7 7 = 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 + ⋯ + ( 6 k − 1 ) = k ( 6 k + 1 ) {\displaystyle 2+5+8+11+\cdots +(6k-1)=k(6k+1)}
2 + 5 + 8 + 11 + ⋯ + ( 6 k − 1 ) ⏟ = k ( 6 k + 1 ) + ( 6 k + 2 ) + ( 6 k + 5 ) = ( k + 1 ) ( 6 k + 7 ) k ( 6 k + 1 ) + ( 6 k + 2 ) + ( 6 k + 5 ) = ( k + 1 ) ( 6 k + 7 ) 6 k 2 + k + 6 k + 2 + 6 k + 5 = 6 k 2 + 7 k + 6 k + 7 0 = 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.
טבעי
