1 2 − 6 7 + 17 12 + 31 20 + ⋯ + ( − 1 ) n − 1 ( 6 n − 1 ) ( 3 n − 2 ) ( 3 n + 1 ) = 1 + ( − 1 ) n + 1 3 n + 1 {\displaystyle {\frac {1}{2}}-{\frac {6}{7}}+{\frac {17}{12}}+{\frac {31}{20}}+\cdots +{\frac {(-1)^{n-1}(6n-1)}{(3n-2)(3n+1)}}=1+{\frac {(-1)^{n+1}}{3n+1}}}
L : ( − 1 ) n − 1 ( 6 n − 1 ) ( 3 n − 2 ) ( 3 n + 1 ) = ( − 1 ) 0 ( 6 ∗ 1 − 1 ) ( 3 ∗ 1 − 2 ) ( 3 ∗ 1 + 1 ) = 5 4 R : 1 + ( − 1 ) n + 1 3 n + 1 = 1 + ( − 1 ) 1 + 1 3 ∗ 1 + 1 = 1 1 4 1.25 = 1.25 {\displaystyle {\begin{aligned}&L:{\frac {(-1)^{n-1}(6n-1)}{(3n-2)(3n+1)}}={\frac {(-1)^{0}(6*1-1)}{(3*1-2)(3*1+1)}}={\frac {5}{4}}\\&R:1+{\frac {(-1)^{n+1}}{3n+1}}=1+{\frac {(-1)^{1+1}}{3*1+1}}=1{\frac {1}{4}}\\&1.25=1.25\\\end{aligned}}}
1 2 − 6 7 + 17 12 + 31 20 + ⋯ + ( − 1 ) k − 1 ( 6 k − 1 ) ( 3 k − 2 ) ( 3 k + 1 ) = 1 + ( − 1 ) k + 1 3 k + 1 {\displaystyle {\frac {1}{2}}-{\frac {6}{7}}+{\frac {17}{12}}+{\frac {31}{20}}+\cdots +{\frac {(-1)^{k-1}(6k-1)}{(3k-2)(3k+1)}}=1+{\frac {(-1)^{k+1}}{3k+1}}}
1 2 − 6 7 + 17 12 + 31 20 + ⋯ + ( − 1 ) k − 1 ( 6 k − 1 ) ( 3 k − 2 ) ( 3 k + 1 ) ⏟ = 1 + ( − 1 ) k + 1 3 k + 1 + ( − 1 ) k ( 6 k + 5 ) ( 3 k + 1 ) ( 3 k + 4 ) = 1 + ( − 1 ) k + 2 3 k + 4 1 + ( − 1 ) k + 1 3 k + 1 + ( − 1 ) k ( 6 k + 5 ) ( 3 k + 1 ) ( 3 k + 4 ) = 1 + ( − 1 ) k + 2 3 k + 4 ( − 1 ) k + 1 ( 3 k + 4 ) + ( − 1 ) k ( 6 k + 5 ) = ( − 1 ) k + 2 ( 3 k + 1 ) ( − 1 ) k ∗ ( − 1 ) ( 3 k + 4 ) + ( − 1 ) k ( 6 k + 5 ) = ( − 1 ) k ∗ ( − 1 ) 2 ( 3 k + 1 ) − 3 k − 4 + 6 k + 5 = 3 k + 1 3 k + 1 = 3 k + 1 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {{\frac {1}{2}}-{\frac {6}{7}}+{\frac {17}{12}}+{\frac {31}{20}}+\cdots +{\frac {(-1)^{k-1}(6k-1)}{(3k-2)(3k+1)}}} _{=1+{\frac {(-1)^{k+1}}{3k+1}}}+{\frac {(-1)^{k}(6k+5)}{(3k+1)(3k+4)}}=1+{\frac {(-1)^{k+2}}{3k+4}}\\&1+{\frac {(-1)^{k+1}}{3k+1}}+{\frac {(-1)^{k}(6k+5)}{(3k+1)(3k+4)}}=1+{\frac {(-1)^{k+2}}{3k+4}}\\&(-1)^{k+1}(3k+4)+(-1)^{k}(6k+5)=(-1)^{k+2}(3k+1)\\&(-1)^{k}*(-1)(3k+4)+(-1)^{k}(6k+5)=(-1)^{k}*(-1)^{2}(3k+1)\\&-3k-4+6k+5=3k+1\\&3k+1=3k+1\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
