1 ∗ 2 ∗ 3 + 4 ∗ 5 ∗ 6 + ⋯ + ( 3 n − 2 ) ( 3 n − 1 ) 3 n = 3 4 n ( n + 1 ) ( 3 n − 2 ) ( 3 n + 1 ) {\displaystyle 1*2*3+4*5*6+\cdots +(3n-2)(3n-1)3n={\frac {3}{4}}n(n+1)(3n-2)(3n+1)}
L : + ( 3 n − 2 ) ( 3 n − 1 ) 3 n → 1 ∗ 2 ∗ 3 = 6 R : 3 4 n ( n + 1 ) ( 3 n − 2 ) ( 3 n + 1 ) → 3 ∗ 2 ∗ 4 4 = 6 6 = 6 √ {\displaystyle {\begin{aligned}&L:+(3n-2)(3n-1)3n\rightarrow 1*2*3=6\\&R:{\frac {3}{4}}n(n+1)(3n-2)(3n+1)\rightarrow {\frac {3*2*4}{4}}=6\\&6=6\surd \\\end{aligned}}}
1 ∗ 2 ∗ 3 + 4 ∗ 5 ∗ 6 + ⋯ + ( 3 k − 2 ) ( 3 k − 1 ) 3 k = 3 4 k ( k + 1 ) ( 3 k − 2 ) ( 3 k + 1 ) {\displaystyle 1*2*3+4*5*6+\cdots +(3k-2)(3k-1)3k={\frac {3}{4}}k(k+1)(3k-2)(3k+1)}
1 ∗ 2 ∗ 3 + 4 ∗ 5 ∗ 6 + ⋯ + ( 3 n − 2 ) ( 3 n − 1 ) 3 n ⏟ = 3 4 k ( k + 1 ) ( 3 k − 2 ) ( 3 k + 1 ) + ( 3 k + 1 ) ( 3 k + 2 ) ( 3 k + 3 ) = 3 4 ( k + 1 ) ( k + 2 ) ( 3 k + 1 ) ( 3 k + 4 ) / ∗ 4 ( k + 1 ) ( 3 k + 1 ) 3 k ( 3 k − 2 ) + 12 ( 3 k + 2 ) = 3 ( k + 2 ) ( 3 k + 4 ) 9 k 2 − 6 k + 36 k + 24 = 3 ( 3 k 2 + 4 k + 6 k + 8 ) 9 k 2 + 30 k + 24 = 9 K 2 + 30 k + 24 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {1*2*3+4*5*6+\cdots +(3n-2)(3n-1)3n} _{={\frac {3}{4}}k(k+1)(3k-2)(3k+1)}+{\color {red}(3k+1)(3k+2)(3k+3)}={\frac {3}{4}}(k+1)(k+2)(3k+1)(3k+4)/*{\frac {4}{(k+1)(3k+1)}}\\&3k(3k-2)+12(3k+2)=3(k+2)(3k+4)\\&9k^{2}-6k+36k+24=3(3k^{2}+4k+6k+8)\\&9k^{2}+30k+24=9K^{2}+30k+24\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
