2 2 ∗ 4 + 2 3 ∗ 5 + 1 4 ∗ 6 + ⋯ + 2 ( n + 1 ) ( n + 3 ) = 5 6 − 2 n + 5 ( n + 2 ) ( n + 3 ) {\displaystyle {\frac {2}{2*4}}+{\frac {2}{3*5}}+{\frac {1}{4*6}}+\cdots +{\frac {2}{(n+1)(n+3)}}={\frac {5}{6}}-{\frac {2n+5}{(n+2)(n+3)}}}
L : 2 ( n + 1 ) ( n + 3 = 2 2 ∗ 4 = 1 4 R : 5 6 − 2 n + 5 ( n + 2 ) ( n + 3 ) = 5 6 − 2 + 5 3 / 4 = 1 4 1 4 = 1 4 √ {\displaystyle {\begin{aligned}&L:{\frac {2}{(n+1)(n+3}}={\frac {2}{2*4}}={\frac {1}{4}}\\&R:{\frac {5}{6}}-{\frac {2n+5}{(n+2)(n+3)}}={\frac {5}{6}}-{\frac {2+5}{3/4}}={\frac {1}{4}}\\&{\frac {1}{4}}={\frac {1}{4}}\surd \\\end{aligned}}}
2 2 ∗ 4 + 2 3 ∗ 5 + 1 4 ∗ 6 + ⋯ + 2 ( k + 1 ) ( k + 3 ) = 5 6 − 2 k + 5 ( k + 2 ) ( k + 3 ) {\displaystyle {\frac {2}{2*4}}+{\frac {2}{3*5}}+{\frac {1}{4*6}}+\cdots +{\frac {2}{(k+1)(k+3)}}={\frac {5}{6}}-{\frac {2k+5}{(k+2)(k+3)}}}
2 2 ∗ 4 + 2 3 ∗ 5 + 1 4 ∗ 6 + ⋯ + 2 ( k + 1 ) ( k + 3 ) ⏟ = 5 6 − 2 k + 5 ( k + 2 ) ( k + 3 ) + 2 ( k + 2 ) ( k + 4 ) = 5 6 − ( 2 k + 7 ) ( k + 3 ) ( k + 4 ) 5 6 − 2 k + 5 ( k + 2 ) ( k + 3 ) + 2 ( k + 2 ) ( k + 4 ) = 5 6 − ( 2 k + 7 ) ( k + 3 ) ( k + 4 ) 2 k + 5 ( k + 2 ) ( k + 3 ) + 2 ( k + 2 ) ( k + 4 ) = − ( 2 k + 7 ) ( k + 3 ) ( k + 4 ) − ( k + 4 ) ( 2 k + 5 ) + 2 k + 6 = − ( k + 2 ) ( 2 k + 7 ) − ( 2 k 2 + 5 k + 8 k + 20 ) + 2 k + 6 = − ( 2 k 2 + 7 k + 4 k + 14 ) − 2 k 2 − 13 k − 20 + 2 k + 6 = − 2 k 2 − 11 k − 14 − 2 k 2 − 11 k − 14 = − 2 k 2 − 11 k − 14 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {{\frac {2}{2*4}}+{\frac {2}{3*5}}+{\frac {1}{4*6}}+\cdots +{\frac {2}{(k+1)(k+3)}}} _{={\frac {5}{6}}-{\frac {2k+5}{(k+2)(k+3)}}}+{\frac {2}{(k+2)(k+4)}}={\frac {5}{6}}-{\frac {(2k+7)}{(k+3)(k+4)}}\\&{\frac {5}{6}}-{\frac {2k+5}{(k+2)(k+3)}}+{\frac {2}{(k+2)(k+4)}}={\frac {5}{6}}-{\frac {(2k+7)}{(k+3)(k+4)}}\\&{\frac {2k+5}{(k+2)(k+3)}}+{\frac {2}{(k+2)(k+4)}}=-{\frac {(2k+7)}{(k+3)(k+4)}}\\&-(k+4)(2k+5)+2k+6=-(k+2)(2k+7)\\&-(2k^{2}+5k+8k+20)+2k+6=-(2k^{2}+7k+4k+14)\\&-2k^{2}-13k-20+2k+6=-2k^{2}-11k-14\\&-2k^{2}-11k-14=-2k^{2}-11k-14\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
