1 + 6 + 11 + 16 + 21 + ⋯ + ( 10 n − 4 ) = n ( 10 n − 3 ) {\displaystyle 1+6+11+16+21+\cdots +(10n-4)=n(10n-3)}
L : 10 − 4 n = 10 − 4 = 6 → 1 + 6 = 7 R : n ( 10 n − 3 ) = 1 ( 10 ∗ 1 − 3 ) = 7 7 = 7 / / {\displaystyle {\begin{aligned}&L:10-4n=10-4=6\rightarrow 1+6=7\\&R:n(10n-3)=1(10*1-3)=7\\&7=7//\end{aligned}}}
1 + 6 + 11 + 16 + 21 + ⋯ + ( 10 k − 4 ) = k ( 10 k − 3 ) {\displaystyle 1+6+11+16+21+\cdots +(10k-4)=k(10k-3)}
1 + 6 + 11 + 16 + 21 + ⋯ + ( 10 k − 4 ) ⏟ = k ( 10 k − 3 ) + ( 10 k + 1 ) ( 10 k + 6 ) = ( k + 1 ) ( 10 k + 7 ) k ( 10 k − 3 ) + ( 10 k + 1 ) ( 10 k + 6 ) = ( k + 1 ) ( 10 k + 7 ) k ( 10 k − 3 ) + ( 10 k + 1 ) + ( 10 k + 6 ) = ( k + 1 ) ( 10 k + 7 ) 10 k 2 − 3 k + 10 k + 1 + 10 k + 6 = 10 k 2 + 7 k + 10 k + 7 10 k 2 + 17 k + 7 = 10 k 2 + 17 k + 7 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {1+6+11+16+21+\cdots +(10k-4)} _{=k(10k-3)}+(10k+1)(10k+6)=(k+1)(10k+7)\\&k(10k-3)+(10k+1)(10k+6)=(k+1)(10k+7)\\&k(10k-3)+(10k+1)+(10k+6)=(k+1)(10k+7)\\&10k^{2}-3k+10k+1+10k+6=10k^{2}+7k+10k+7\\&10k^{2}+17k+7=10k^{2}+17k+7\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
