4 + 8 5 + 12 25 + ⋯ + 4 n 5 n − 1 = 1 4 ( 25 − 4 n + 5 5 n − 1 {\displaystyle 4+{\frac {8}{5}}+{\frac {12}{25}}+\cdots +{\frac {4n}{5^{n-1}}}={\frac {1}{4}}(25-{\frac {4n+5}{5^{n-1}}}}
L : 4 n 5 n − 1 = 4 5 0 = 4 R : 1 4 ∗ ( 25 − 4 n + 5 5 n − 1 = 1 4 ∗ ( 25 − 4 + 5 5 0 ) = 4 4 = 4 √ {\displaystyle {\begin{aligned}&L:{\frac {4n}{5^{n-1}}}={\frac {4}{5^{0}}}=4\\&R:{\frac {1}{4}}*(25-{\frac {4n+5}{5^{n-1}}}={\frac {1}{4}}*(25-{\frac {4+5}{5^{0}}})=4\\&4=4\surd \\\end{aligned}}}
4 + 8 5 + 12 25 + ⋯ + 4 k 5 k − 1 = 1 4 ( 25 − 4 k + 5 5 k − 1 {\displaystyle 4+{\frac {8}{5}}+{\frac {12}{25}}+\cdots +{\frac {4k}{5^{k-1}}}={\frac {1}{4}}(25-{\frac {4k+5}{5^{k-1}}}}
4 + 8 5 + 12 25 + ⋯ + 4 k 5 k − 1 ⏟ = 1 4 ( 25 − 4 k + 5 5 k − 1 + 4 k + 4 5 k = 1 4 ( 25 − 4 k + 9 5 k 1 4 ( 25 − 4 k + 5 5 k − 1 ) + 4 k + 4 5 k = 1 4 ( 25 − 4 k + 9 5 k ) 25 − 4 k + 5 5 k − 1 + 4 k + 4 5 k ∗ 4 = 25 − 4 k + 9 5 k − 4 k − 5 5 k − 1 + 16 k + 16 5 k = − 4 k − 9 5 k 5 ( − 4 k − 5 ) + 16 k + 16 = − 4 k − 9 − 20 k − 25 + 16 k + 16 = − 4 k − 9 − 4 k − 9 = − 4 k − 9 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {4+{\frac {8}{5}}+{\frac {12}{25}}+\cdots +{\frac {4k}{5^{k-1}}}} _{={\frac {1}{4}}(25-{\frac {4k+5}{5^{k-1}}}}+{\frac {4k+4}{5^{k}}}={\frac {1}{4}}(25-{\frac {4k+9}{5^{k}}}\\&{\frac {1}{4}}(25-{\frac {4k+5}{5^{k-1}}})+{\frac {4k+4}{5^{k}}}={\frac {1}{4}}(25-{\frac {4k+9}{5^{k}}})\\&25-{\frac {4k+5}{5^{k-1}}}+{\frac {4k+4}{5^{k}}}*4=25-{\frac {4k+9}{5^{k}}}\\&{\frac {-4k-5}{5^{k-1}}}+{\frac {16k+16}{5^{k}}}={\frac {-4k-9}{5^{k}}}\\&5(-4k-5)+16k+16=-4k-9\\&-20k-25+16k+16=-4k-9\\&-4k-9=-4k-9\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
