2 2 − 4 2 + 5 2 − 7 2 + ⋯ + ( 3 n + 1 ) 2 = − 6 n ( n + 1 ) {\displaystyle 2^{2}-4^{2}+5^{2}-7^{2}+\cdots +(3n+1)^{2}=-6n(n+1)}
L : ( 3 n + 1 ) 2 = ( 3 + 1 ) 2 = 4 2 → 2 2 − 4 2 = − 12 R : − 6 n ( n + 1 ) = − 6 ( 1 + 1 ) = − 12 − 12 = − 12 {\displaystyle {\begin{aligned}&L:(3n+1)^{2}=(3+1)^{2}=4^{2}\rightarrow 2^{2}-4^{2}=-12\\&R:-6n(n+1)=-6(1+1)=-12\\&-12=-12\\\end{aligned}}}
2 2 − 4 2 + 5 2 − 7 2 + ⋯ + ( 3 k + 1 ) 2 = − 6 k ( k + 1 ) {\displaystyle 2^{2}-4^{2}+5^{2}-7^{2}+\cdots +(3k+1)^{2}=-6k(k+1)}
2 2 − 4 2 + 5 2 − 7 2 + ⋯ + ( 3 k + 1 ) 2 ⏟ = − 6 k ( k + 1 ) + ( 3 k + 2 ) 2 − ( 3 k + 4 ) 2 = − ( 6 k + 6 ) ( k + 2 ) − 6 k ( k + 1 ) + ( 3 k + 2 ) 2 − ( 3 k + 4 ) 2 = − ( 6 k + 6 ) ( k + 2 ) − 6 k 2 − 6 k + 9 k 2 + 12 k + 4 − 9 k 2 − 24 k + 16 = − 6 k 2 − 18 k − 12 − 6 k 2 − 18 k − 12 = − 6 k 2 − 18 k − 12 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {2^{2}-4^{2}+5^{2}-7^{2}+\cdots +(3k+1)^{2}} _{=-6k(k+1)}+(3k+2)^{2}-(3k+4)^{2}=-(6k+6)(k+2)\\&-6k(k+1)+(3k+2)^{2}-(3k+4)^{2}=-(6k+6)(k+2)\\&-6k^{2}-6k+9k^{2}+12k+4-9k^{2}-24k+16=-6k^{2}-18k-12\\&-6k^{2}-18k-12=-6k^{2}-18k-12\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
