1 ∗ 1 ! + 2 ∗ 2 + 3 ∗ 3 ! + ⋯ + n ∗ n ! = ( n + 1 ) ! − 1 {\displaystyle 1*1!+2*2+3*3!+\cdots +n*n!=(n+1)!-1}
L : n ∗ n ! = 1 ∗ 1 ! R : ( n + 1 ) ! − 1 = ( 2 ) ! − 1 = 1 1 = 1 √ {\displaystyle {\begin{aligned}&L:n*n!=1*1!\\&R:(n+1)!-1=(2)!-1=1\\&1=1\surd \\\end{aligned}}}
1 ∗ 1 ! + 2 ∗ 2 + 3 ∗ 3 ! + ⋯ + k ∗ k ! = ( k + 1 ) ! − 1 {\displaystyle 1*1!+2*2+3*3!+\cdots +k*k!=(k+1)!-1}
1 ∗ 1 ! + 2 ∗ 2 + 3 ∗ 3 ! + ⋯ + k ∗ k ! ⏟ = ( k + 1 ) ! − 1 + ( k + 1 ) ( k + 1 ) ! = ( k + 2 ) ! − 1 ( k + 1 ) ! − 1 + ( k + 1 ) ( k + 1 ) ! = ( k + 2 ) ! ⏟ 1 ∗ 2 ∗ 3 ∗ ⋯ ∗ k ∗ ( k + 1 ) ⏟ ( k + 1 ) ! ∗ ( k + 2 ) − 1 ( k + 1 ) ! + ( k + 1 ) ( k + 1 ) ! = ( k + 2 ) ∗ ( k + 1 ) ! / : ( k + 1 ) ! 1 + ( k + 1 ) = ( k + 2 ) k + 2 = k + 2 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {1*1!+2*2+3*3!+\cdots +k*k!} _{=(k+1)!-1}+(k+1)(k+1)!=(k+2)!-1\\&(k+1)!-1+(k+1)(k+1)!=\underbrace {(k+2)!} _{{\color {blue}\underbrace {1*2*3*\cdots *k*(k+1)} _{(k+1)!}}*(k+2)}-1\\&(k+1)!+(k+1)(k+1)!=(k+2)*(k+1)!/:(k+1)!\\&1+(k+1)=(k+2)\\&k+2=k+2\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
