− 3 + 7 − 11 + 15 − ⋯ + ( − 1 ) n ( 4 n − 1 ) = 1 2 [ ( − n ) n ( 4 n + 1 ) − 1 ] {\displaystyle -3+7-11+15-\cdots +(-1)^{n}(4n-1)={\frac {1}{2}}[(-n)^{n}(4n+1)-1]}
L : ( − 1 ) n ( 4 n − 1 ) = ( − 1 ) ( 4 − 1 ) = − 3 R : 1 2 [ ( − n ) n ( 4 n + 1 ) − 1 ] = 1 2 [ ( − 1 ) ( 4 + 1 ) − 1 ] = − 6 2 = − 3 − 3 = − 3 {\displaystyle {\begin{aligned}&L:(-1)^{n}(4n-1)=(-1)(4-1)=-3\\&R:{\frac {1}{2}}[(-n)^{n}(4n+1)-1]={\frac {1}{2}}[(-1)(4+1)-1]={\frac {-6}{2}}=-3\\&-3=-3\\\end{aligned}}}
− 3 + 7 − 11 + 15 − ⋯ + ( − 1 ) k ( 4 k − 1 ) = 1 2 [ ( − k ) k ( 4 k + 1 ) − 1 ] {\displaystyle -3+7-11+15-\cdots +(-1)^{k}(4k-1)={\frac {1}{2}}[(-k)^{k}(4k+1)-1]}
− 3 + 7 − 11 + 15 − ⋯ + ( − 1 ) k ( 4 k − 1 ) ⏟ = 1 2 [ ( − k ) k ( 4 k + 1 ) − 1 ] + ( − 1 ) k + 1 ( 4 k + 3 ) = 1 2 [ ( − 1 ) k + 1 ( 4 k + 5 ) − 1 ] 1 2 [ ( − 1 ) k ( 4 k + 1 ) − 1 ] + ( − 1 ) k + 1 ( 4 k + 3 ) = 1 2 [ ( − 1 ) k + 1 ( 4 k + 5 ) − 1 ] ( − 1 ) k ( 4 k + 1 ) − 1 + 2 ( − 1 ) k + 1 ( 4 k + 3 ) = ( − 1 ) k + 1 ( 4 k + 5 ) − 1 − ( − 1 ) k + 1 ( 4 k + 1 ) + 2 ( − 1 ) k + 1 94 k + 3 ) = ( − 1 ) k + 1 ( 4 k + 5 ) − ( 4 k + 1 ) + 2 ( 4 k + 3 ) = ( 4 k + 5 ) − 4 k − 1 + 8 k + 6 = 4 k + 5 4 k + 5 = 4 k + 5 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {-3+7-11+15-\cdots +(-1)^{k}(4k-1)} _{={\frac {1}{2}}[(-k)^{k}(4k+1)-1]}+(-1)^{k+1}(4k+3)={\frac {1}{2}}[(-1)^{k+1}(4k+5)-1]\\&{\frac {1}{2}}[(-1)^{k}(4k+1)-1]+(-1)^{k+1}(4k+3)={\frac {1}{2}}[(-1)^{k+1}(4k+5)-1]\\&(-1)^{k}(4k+1)-1+2(-1)^{k+1}(4k+3)=(-1)^{k+1}(4k+5)-1\\&-(-1)^{k+1}(4k+1)+2(-1)^{k+1}94k+3)=(-1)^{k+1}(4k+5)\\&-(4k+1)+2(4k+3)=(4k+5)\\&-4k-1+8k+6=4k+5\\&4k+5=4k+5\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
![{\displaystyle -3+7-11+15-\cdots +(-1)^{n}(4n-1)={\frac {1}{2}}[(-n)^{n}(4n+1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e81cbab2b4288388f2971eea399378e1d8816cb)
![{\displaystyle {\begin{aligned}&L:(-1)^{n}(4n-1)=(-1)(4-1)=-3\\&R:{\frac {1}{2}}[(-n)^{n}(4n+1)-1]={\frac {1}{2}}[(-1)(4+1)-1]={\frac {-6}{2}}=-3\\&-3=-3\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89875336a2abb4db212073b77409eceedbc3e31c)
טבעי![{\displaystyle -3+7-11+15-\cdots +(-1)^{k}(4k-1)={\frac {1}{2}}[(-k)^{k}(4k+1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1bdd74a796b3d446142991b9a0bb8d6383a38c)
![{\displaystyle {\begin{aligned}&\underbrace {-3+7-11+15-\cdots +(-1)^{k}(4k-1)} _{={\frac {1}{2}}[(-k)^{k}(4k+1)-1]}+(-1)^{k+1}(4k+3)={\frac {1}{2}}[(-1)^{k+1}(4k+5)-1]\\&{\frac {1}{2}}[(-1)^{k}(4k+1)-1]+(-1)^{k+1}(4k+3)={\frac {1}{2}}[(-1)^{k+1}(4k+5)-1]\\&(-1)^{k}(4k+1)-1+2(-1)^{k+1}(4k+3)=(-1)^{k+1}(4k+5)-1\\&-(-1)^{k+1}(4k+1)+2(-1)^{k+1}94k+3)=(-1)^{k+1}(4k+5)\\&-(4k+1)+2(4k+3)=(4k+5)\\&-4k-1+8k+6=4k+5\\&4k+5=4k+5\\&0=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0217278d70977dabe78a1813da5cf9539242c5)
