3 + 7 + 11 + 15 + 19 + 23 + ⋯ + ( 12 n − 1 ) = 3 n ( 6 n + 1 ) {\displaystyle 3+7+11+15+19+23+\cdots +(12n-1)=3n(6n+1)}
L : ( 12 n − 1 ) = 12 − 1 = 11 → 3 + 7 + 11 = 21 R : 3 n ( 6 n + 1 ) = 3 ( 6 + 1 ) = 21 21 = 21 {\displaystyle {\begin{aligned}&L:(12n-1)=12-1=11\rightarrow 3+7+11=21\\&R:3n(6n+1)=3(6+1)=21\\&21=21\\\end{aligned}}}
3 + 7 + 11 + 15 + 19 + 23 + ⋯ + ( 12 k − 1 ) = 3 k ( 6 k + 1 ) {\displaystyle 3+7+11+15+19+23+\cdots +(12k-1)=3k(6k+1)}
3 + 7 + 11 + 15 + 19 + 23 + ⋯ + ( 12 k − 1 ) ⏟ = 3 k ( 6 k + 1 ) + ( 12 k + 3 ) + ( 12 k + 7 ) + ( 12 k + 11 ) = ( 3 k + 3 ) ( 6 k + 7 ) 3 k ( 6 k + 1 ) + ( 12 k + 3 ) + ( 12 k + 7 ) + ( 12 k + 11 ) = ( 3 k + 3 ) ( 6 k + 7 ) 18 k 2 + 3 k + 12 k + 3 + 12 k + 7 + 12 k + 11 = 3 ( 6 k 2 + 7 k + 6 k + 7 ) 18 k 2 + 39 k + 21 = 3 ( 6 k 2 + 13 k + 7 ) 6 k 2 + 13 k + 7 = 6 k 2 + 13 k + 7 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {3+7+11+15+19+23+\cdots +(12k-1)} _{=3k(6k+1)}+(12k+3)+(12k+7)+(12k+11)=(3k+3)(6k+7)\\&3k(6k+1)+(12k+3)+(12k+7)+(12k+11)=(3k+3)(6k+7)\\&18k^{2}+3k+12k+3+12k+7+12k+11=3(6k^{2}+7k+6k+7)\\&18k^{2}+39k+21=3(6k^{2}+13k+7)\\&6k^{2}+13k+7=6k^{2}+13k+7\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
