( 1 1 + 1 1 ) ( 1 2 + 1 4 ) ( 1 3 + 1 9 ) ∗ ⋯ ∗ ( 1 n + 1 n 2 ) = n + 1 n ! {\displaystyle ({\frac {1}{1}}+{\frac {1}{1}})({\frac {1}{2}}+{\frac {1}{4}})({\frac {1}{3}}+{\frac {1}{9}})*\cdots *({\frac {1}{n}}+{\frac {1}{n^{2}}})={\frac {n+1}{n!}}}
L : 1 n + 1 n 2 = 1 1 + 1 1 2 = 2 R : n + 1 n ! = 1 + 1 1 ! = 1 + 1 1 = 2 2 = 2 {\displaystyle {\begin{aligned}L:{\frac {1}{n}}+{\frac {1}{n^{2}}}={\frac {1}{1}}+{\frac {1}{1^{2}}}=2\\R:{\frac {n+1}{n!}}={\frac {1+1}{1!}}={\frac {1+1}{1}}=2\\2=2\\\end{aligned}}}
( 1 1 + 1 1 ) ( 1 2 + 1 4 ) ( 1 3 + 1 9 ) ∗ ⋯ ∗ ( 1 k + 1 k 2 ) = k + 1 k ! {\displaystyle ({\frac {1}{1}}+{\frac {1}{1}})({\frac {1}{2}}+{\frac {1}{4}})({\frac {1}{3}}+{\frac {1}{9}})*\cdots *({\frac {1}{k}}+{\frac {1}{k^{2}}})={\frac {k+1}{k!}}}
( 1 1 + 1 1 ) ( 1 2 + 1 4 ) ( 1 3 + 1 9 ) ∗ ⋯ ∗ ( 1 k + 1 k 2 ) ∗ ( 1 k + 1 + 1 ( k + 1 ) 2 ) = k + 1 + 1 ( k + 1 ) ! ( 1 1 + 1 1 ) ( 1 2 + 1 4 ) ( 1 3 + 1 9 ) ∗ ⋯ ∗ ( 1 k + 1 k 2 ) ⏟ = k + 1 k ! ∗ ( 1 k + 1 + 1 ( k + 1 ) 2 ) = k + 1 + 1 ( k + 1 ) ! k + 1 k ! ∗ ( 1 k + 1 + 1 ( k + 1 ) 2 ) = k + 2 ( k + 1 ) ! k + 1 k ! ∗ ( 1 k + 1 + 1 ( k + 1 ) 2 ) = k + 2 ( k + 1 ) ! k + 1 k ! ∗ ( k + 1 + 1 ( k + 1 ) 2 ) = k + 2 ( k + 1 ) ! k + 1 k ! ∗ k + 2 ( k + 1 ) 2 = k + 2 ( k + 1 ) ! 1 k ! ∗ k + 2 ( k + 1 ) = k + 2 ( k + 1 ) ! k ! = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ k ( k + 1 ) ! = 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ k ∗ ( k + 1 ) k + 2 ( k + 1 ) ! = k + 2 ( k + 1 ) ! {\displaystyle {\begin{aligned}&({\frac {1}{1}}+{\frac {1}{1}})({\frac {1}{2}}+{\frac {1}{4}})({\frac {1}{3}}+{\frac {1}{9}})*\cdots *({\frac {1}{k}}+{\frac {1}{k^{2}}})*({\frac {1}{k+1}}+{\frac {1}{(k+1)^{2}}})={\frac {k+1+1}{(k+1)!}}\\&\underbrace {({\frac {1}{1}}+{\frac {1}{1}})({\frac {1}{2}}+{\frac {1}{4}})({\frac {1}{3}}+{\frac {1}{9}})*\cdots *({\frac {1}{k}}+{\frac {1}{k^{2}}})} _{={\frac {k+1}{k!}}}*({\frac {1}{k+1}}+{\frac {1}{(k+1)^{2}}})={\frac {k+1+1}{(k+1)!}}\\&{\frac {k+1}{k!}}*({\frac {1}{k+1}}+{\frac {1}{(k+1)^{2}}})={\frac {k+2}{(k+1)!}}\\&{\frac {k+1}{k!}}*({\frac {1}{k+1}}+{\frac {1}{(k+1)^{2}}})={\frac {k+2}{(k+1)!}}\\&{\frac {k+1}{k!}}*({\frac {k+1+1}{(k+1)^{2}}})={\frac {k+2}{(k+1)!}}\\&{\frac {k+1}{k!}}*{\frac {k+2}{(k+1)^{2}}}={\frac {k+2}{(k+1)!}}\\&{\frac {1}{k!}}*{\frac {k+2}{(k+1)}}={\frac {k+2}{(k+1)!}}\\&k!=1*2*3*\cdots *k\\&(k+1)!=1*2*3*\cdots *k*(k+1)\\&{\frac {k+2}{(k+1)!}}={\frac {k+2}{(k+1)!}}\\\end{aligned}}}
האידוקציה נכון עבור כל n {\displaystyle \ n} טבעי החל מ n = 1 {\displaystyle \ n=1} על פי 3 שלבי האינדוקציה.