( 2 n + 2 ) + ( 2 n + 4 ) + ( 2 n + 6 ) + ⋯ + ( 6 n ) = 2 n ( 4 n + 1 ) {\displaystyle (2n+2)+(2n+4)+(2n+6)+\cdots +(6n)=2n(4n+1)}
L : ( 6 n ) = 6 → ( 2 + 2 ) ( 2 + 4 ) = 10 R : 2 n ( 4 n + 1 ) = 2 ( 4 + 1 ) = 10 10 = 10 {\displaystyle {\begin{aligned}&L:(6n)=6\rightarrow (2+2)(2+4)=10\\&R:2n(4n+1)=2(4+1)=10\\&10=10\\\end{aligned}}}
( 2 k + 2 ) + ( 2 k + 4 ) + ( 2 k + 6 ) + ⋯ + ( 6 k ) = 2 k ( 4 k + 1 ) {\displaystyle (2k+2)+(2k+4)+(2k+6)+\cdots +(6k)=2k(4k+1)}
( 2 k + 4 ) + ( 2 k + 6 ) + ⋯ + ( 6 k ) ⏟ = 2 k ( 4 k + 1 ) − ( 2 k + 2 ) + ( 6 k + 2 ) + ( 6 k + 4 ) + ( 6 k + 6 ) = ( 2 k + 2 ) ( 4 k + 5 ) 2 k ( 4 k + 1 ) − ( 2 k + 2 ) + ( 6 k + 2 ) + ( 6 k + 4 ) + ( 6 k + 6 ) = ( 2 k + 2 ) ( 4 k + 5 ) 8 k 2 + 2 k − 2 k − 2 + 18 k + 12 = 8 k 2 + 18 k + 10 8 k 2 + 18 k + 10 = 8 k 2 + 18 k + 10 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {(2k+4)+(2k+6)+\cdots +(6k)} _{=2k(4k+1)-(2k+2)}+(6k+2)+(6k+4)+(6k+6)=(2k+2)(4k+5)\\&2k(4k+1)-(2k+2)+(6k+2)+(6k+4)+(6k+6)=(2k+2)(4k+5)\\&8k^{2}+2k-2k-2+18k+12=8k^{2}+18k+10\\&8k^{2}+18k+10=8k^{2}+18k+10\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
