12−22+32−42+⋯+(2n−1)2−(2n)2=−n(2n+1){\displaystyle 1^{2}-2^{2}+3^{2}-4^{2}+\cdots +(2n-1)^{2}-(2n)^{2}=-n(2n+1)}
L:(2n−1)2−(2n)2=(2−1)2−(2)2=12−22→1−4=−3R:−n(2n+1)=−(2+1)=−3−3=−3{\displaystyle {\begin{aligned}&L:(2n-1)^{2}-(2n)^{2}=(2-1)^{2}-(2)^{2}=1^{2}-2^{2}\rightarrow 1-4=-3\\&R:-n(2n+1)=-(2+1)=-3\\&-3=-3\\\end{aligned}}}
12−22+32−42+⋯+(2k−1)2−(2k)2=−k(2k+1){\displaystyle 1^{2}-2^{2}+3^{2}-4^{2}+\cdots +(2k-1)^{2}-(2k)^{2}=-k(2k+1)}
12−22+32−42+⋯+(2k−1)2−(2k)2⏟=−k(2k+1)+(2k+1)2−(2k+2)2=−(k+1)(2k+3)−k(2k+1)+(2k+1)2−(2k+2)2=−(k+1)(2k+3)−2k2−k+4k2+4k+1−4k2−8k−4=−2k2−5k−3−2k2−5k−3=−2k2−5k−30=0{\displaystyle {\begin{aligned}&\underbrace {1^{2}-2^{2}+3^{2}-4^{2}+\cdots +(2k-1)^{2}-(2k)^{2}} _{=-k(2k+1)}+(2k+1)^{2}-(2k+2)^{2}=-(k+1)(2k+3)\\&-k(2k+1)+(2k+1)^{2}-(2k+2)^{2}=-(k+1)(2k+3)\\&-2k^{2}-k+4k^{2}+4k+1-4k^{2}-8k-4=-2k^{2}-5k-3\\&-2k^{2}-5k-3=-2k^{2}-5k-3\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.