2 + 6 + 10 + 14 + ⋯ + 4 ( 4 n + 6 ) = 2 ( n + 2 ) 2 {\displaystyle 2+6+10+14+\cdots +4(4n+6)=2(n+2)^{2}}
L : 4 ( 4 n + 6 ) = 1 ( 1 ∗ 4 + 6 ) = 10 → 2 + 6 + 10 = 18 R : 2 ( n + 2 ) 2 = 2 ( 1 + 2 ) 2 = 18 18 = 18 {\displaystyle {\begin{aligned}&L:4(4n+6)=1(1*4+6)=10\rightarrow 2+6+10=18\\&R:2(n+2)^{2}=2(1+2)^{2}=18\\&18=18\end{aligned}}}
2 + 6 + 10 + 14 + ⋯ + 4 ( 4 k + 6 ) = 2 ( k + 2 ) 2 {\displaystyle 2+6+10+14+\cdots +4(4k+6)=2(k+2)^{2}}
2 + 6 + 10 + 14 + ⋯ + 4 ( 4 k + 6 ) ⏟ = 2 ( k + 2 ) 2 + ( 4 k + 10 ) = 2 ( k + 3 ) 2 2 ( k + 2 ) 2 + ( 4 k + 10 ) = 2 ( k + 3 ) 2 2 ( k + 2 ) 2 + 2 ( 2 k + 5 ) = 2 ( k + 3 ) 2 k 2 + 4 k + 4 + 2 k + 5 = k 2 + 6 k + 9 k 2 + 6 k + 9 = k 2 + 6 k + 9 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {2+6+10+14+\cdots +4(4k+6)} _{=2(k+2)^{2}}+(4k+10)=2(k+3)^{2}\\&2(k+2)^{2}+(4k+10)=2(k+3)^{2}\\&2(k+2)^{2}+2(2k+5)=2(k+3)^{2}\\&k^{2}+4k+4+2k+5=k^{2}+6k+9\\&k^{2}+6k+9=k^{2}+6k+9\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.