5 + 25 + 125 + ⋯ + 5 n = 5 4 ( 5 n − 1 ) {\displaystyle 5+25+125+\cdots +5^{n}={\frac {5}{4}}(5n-1)}
L : 5 n = 5 1 = 5 R : 5 4 ( 5 n − 1 ) = 5 4 ( 5 1 − 1 ) = 5 5 = 5 √ {\displaystyle {\begin{aligned}&L:5^{n}=5^{1}=5\\&R:{\frac {5}{4}}(5^{n}-1)={\frac {5}{4}}(5^{1}-1)=5\\&5=5\surd \\\end{aligned}}}
5 + 25 + 125 + ⋯ + 5 k = 5 4 ( 5 k − 1 ) {\displaystyle 5+25+125+\cdots +5^{k}={\frac {5}{4}}(5k-1)}
5 + 25 + 125 + ⋯ + 5 k ⏟ = 5 4 ( 5 k − 1 ) + 5 k + 1 = 5 4 ( 5 k + 1 − 1 ) 5 4 ( 5 k − 1 ) + 5 k + 1 = 5 4 ( 5 k + 1 − 1 ) 5 ( 5 k − 1 ) + 4 ∗ 5 k + 1 = 5 ( 5 k + 1 − 1 ) 5 ∗ 5 k − 5 + 4 ∗ 5 k + 1 = 5 ∗ 5 k + 1 − 5 5 k + 1 + 4 ∗ 5 k + 1 = 5 ∗ 5 k + 1 / : 5 k + 1 1 + 4 = 5 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {5+25+125+\cdots +5^{k}} _{={\frac {5}{4}}(5k-1)}+5^{k+1}={\frac {5}{4}}(5^{k+1}-1)\\&{\frac {5}{4}}(5k-1)+5^{k+1}={\frac {5}{4}}(5^{k+1}-1)\\&5(5^{k}-1)+4*5^{k+1}=5(5^{k+1}-1)\\&5*5^{k}-5+4*5^{k+1}=5*5^{k+1}-5\\&5^{k+1}+4*5^{k+1}=5*5^{k+1}/:5^{k+1}\\&1+4=5\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
