8 + 14 + 20 + 26 + ⋯ + ( 3 n + 2 ) = n 4 ( 3 n + 10 ) {\displaystyle 8+14+20+26+\cdots +(3n+2)={\frac {n}{4}}(3n+10)}
L : 3 n + 2 = 3 + 2 = 5 R : n 4 ( 3 n + 10 ) = 2 4 ( 6 + 10 ) = 8 8 = 8 {\displaystyle {\begin{aligned}&L:3n+2=3+2=5\\&R:{\frac {n}{4}}(3n+10)={\frac {2}{4}}(6+10)=8\\&8=8\\\end{aligned}}}
8 + 14 + 20 + 26 + ⋯ + ( 3 k + 2 ) = k 4 ( 3 k + 10 ) {\displaystyle 8+14+20+26+\cdots +(3k+2)={\frac {k}{4}}(3k+10)}
8 + 14 + 20 + 26 + ⋯ + ( 3 k + 2 ) ⏟ = k 4 ( 3 k + 10 ) + ( 3 k + 8 ) = k + 2 4 ( 3 k + 16 ) k 4 ( 3 k + 10 ) + ( 3 k + 8 ) = k + 2 4 ( 3 k + 16 ) k ( 3 k + 10 ) + 4 ( 3 k + 8 ) = ( k + 2 ) ( 3 k + 16 ) 3 k 2 + 10 k + 12 k + 32 = 3 k 2 + 16 k + 6 k + 32 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {8+14+20+26+\cdots +(3k+2)} _{={\frac {k}{4}}(3k+10)}+(3k+8)={\frac {k+2}{4}}(3k+16)&{\frac {k}{4}}(3k+10)+(3k+8)={\frac {k+2}{4}}(3k+16)\\&k(3k+10)+4(3k+8)=(k+2)(3k+16)\\&3k^{2}+10k+12k+32=3k^{2}+16k+6k+32\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
