1+3+32+⋯+33n−113=Z{\displaystyle {\frac {1+3+3^{2}+\cdots +3^{3n-1}}{13}}=Z}
33n−1=Z33∗1−1=32↓1+3+3213=1+3+913=13131313=Z√{\displaystyle {\begin{aligned}&3^{3n-1}=Z\\&3^{3*1-1}=3^{2}\\&\downarrow \\&{\frac {1+3+3^{2}}{13}}={\frac {1+3+9}{13}}={\frac {13}{13}}\\&{\frac {13}{13}}=Z\surd \\\end{aligned}}}
1+3+32+⋯+33k−113=Z{\displaystyle {\frac {1+3+3^{2}+\cdots +3^{3k-1}}{13}}=Z}
10+31+32+⋯+33K−1+33k+33k+1+33k+213=Z10+31+32+⋯+33K−113⏟=Z+33k+33k+1+33k+213=ZZ+33k+33k+1+33k+213=ZZ=Z√33k+33k+1+33k+213=Z33k(1+31+3213=Z33k(13)13=Z√33k=Z{\displaystyle {\begin{aligned}&{\frac {1^{0}+3^{1}+3^{2}+\cdots +3^{3K-1}+{\color {red}3^{3k}+3^{3k+1}}+3^{3k+2}}{13}}=Z\\&\underbrace {\frac {1^{0}+3^{1}+3^{2}+\cdots +3^{3K-1}}{13}} _{=Z}+{\frac {3^{3k}+3^{3k+1}+3^{3k+2}}{13}}=Z\\&Z+{\frac {3^{3k}+3^{3k+1}+3^{3k+2}}{13}}=Z\\&Z=Z\surd &{\frac {3^{3k}+3^{3k+1}+3^{3k+2}}{13}}=Z\\&{\frac {3^{3k}(1+3^{1}+3^{2}}{13}}=Z\\&{\frac {3^{3k}(13)}{13}}=Z\\&\surd 3^{3k}=Z\\\end{aligned}}}
33k{\displaystyle 3^{3k}} שלם עבור K טבעי.הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.