6 + 20 + 42 + + ⋯ + 2 n ( 2 n + 1 ) = n 3 ( n + 1 ) ( 4 n + 5 ) {\displaystyle 6+20+42++\cdots +2n(2n+1)={\frac {n}{3}}(n+1)(4n+5)}
L : 2 n ( 2 n + 1 ) ⇒ 2 ∗ 1 ( 2 ∗ 1 + 1 ) = 6 ⏟ 6 R : n 3 ( n + 1 ) ( 4 n + 5 ) ⇒ 1 3 ( 1 + 1 ) ( 4 ∗ 1 + 5 ) = 6 6 = 6 √ {\displaystyle {\begin{aligned}L:2n(2n+1)\Rightarrow 2*1(2*1+1)=\underbrace {6} _{6}\\R:{\frac {n}{3}}(n+1)(4n+5)\Rightarrow {\frac {1}{3}}(1+1)(4*1+5)=6\\6=6\surd \\\end{aligned}}}
6 + 20 + 42 + ⋯ + 2 k ( 2 k + 1 ) = k 3 ( k + 1 ) ( 4 k + 5 ) {\displaystyle 6+20+42+\cdots +2k(2k+1)={\frac {k}{3}}(k+1)(4k+5)}
1 ∗ 2 + 3 ∗ 4 + 5 ∗ 6 + ⋯ + 2 k ( 2 k + 1 ) ⏟ = k 3 ( k + 1 ) ( 4 k + 5 ) + [ 2 ( k + 1 ) ] ∗ [ 2 ( k + 1 ) + 1 ] = k + 1 3 ( k + 1 + 1 ) [ 4 ( k + 1 ) + 5 ] k 3 ( k + 1 ) ( 4 k + 5 ) + ( 2 k + 2 ) ( 2 k + 3 ) = k + 1 3 ( k + 2 ) ( 4 k + 9 ) / ∗ 3 k ( k + 1 ) ( 4 k + 5 ) + 3 ∗ 2 ( k + 1 ) ( 2 k + 3 ) = ( k + 1 ) ( k + 2 ) ( 4 k + 9 ) / : ( k + 1 ) k ( 4 k + 5 ) + 3 ∗ 2 ( 2 k + 3 ) = ( k + 2 ) ( 4 k + 9 ) 4 k 2 + 5 k + 12 k + 18 = 4 k 2 + 9 k + 8 k + 18 4 k 2 + 17 k + 18 = 4 k 2 + 17 k + 18 0 = 0 √ {\displaystyle {\begin{aligned}\underbrace {1*2+3*4+5*6+\cdots +2k(2k+1)} _{={\frac {k}{3}}(k+1)(4k+5)}+{\color {red}[2(k+1)]*[2(k+1)+1]}={\color {red}{\frac {k+1}{3}}(k+1+1)[4(k+1)+5]}\\{\frac {k}{3}}(k+1)(4k+5)+(2k+2)(2k+3)={\frac {k+1}{3}}(k+2)(4k+9)/*3\\k(k+1)(4k+5)+3*2(k+1)(2k+3)=(k+1)(k+2)(4k+9)/:(k+1)\\k(4k+5)+3*2(2k+3)=(k+2)(4k+9)\\4k^{2}+5k+12k+18=4k^{2}+9k+8k+18\\4k^{2}+17k+18=4k^{2}+17k+18\\0=0\surd \\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
![{\displaystyle {\begin{aligned}\underbrace {1*2+3*4+5*6+\cdots +2k(2k+1)} _{={\frac {k}{3}}(k+1)(4k+5)}+{\color {red}[2(k+1)]*[2(k+1)+1]}={\color {red}{\frac {k+1}{3}}(k+1+1)[4(k+1)+5]}\\{\frac {k}{3}}(k+1)(4k+5)+(2k+2)(2k+3)={\frac {k+1}{3}}(k+2)(4k+9)/*3\\k(k+1)(4k+5)+3*2(k+1)(2k+3)=(k+1)(k+2)(4k+9)/:(k+1)\\k(4k+5)+3*2(2k+3)=(k+2)(4k+9)\\4k^{2}+5k+12k+18=4k^{2}+9k+8k+18\\4k^{2}+17k+18=4k^{2}+17k+18\\0=0\surd \\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52039ddb9eed1e7c0d46d564c34aafa8ba4125c2)
