2 − 8 + 14 − 20 + 26 − 32 + ⋯ = 1 2 [ ( − 1 ) n + 1 ( 6 n − 1 ) − 1 ] {\displaystyle 2-8+14-20+26-32+\cdots ={\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]}
a 1 = 2 d = 6 a n = a 1 + ( n − 1 ) d ↓ a n = 2 + ( n − 1 ) 6 a n = 2 + 6 n − n a n = 6 n − 4 ↓ ( − 1 ) n + 1 ( 6 n − 4 ) {\displaystyle {\begin{aligned}&a_{1}=2&d=6\\&a_{n}=a_{1}+(n-1)d\\&\downarrow \\&a_{n}=2+(n-1)6\\&a_{n}=2+6n-n\\&a_{n}=6n-4\\&\downarrow \\&(-1)^{n+1}(6n-4)\\\end{aligned}}}
2 − 8 + 14 − 20 + 26 − 32 + ⋯ + ( − 1 ) n + 1 ( 6 n − 4 ) = 1 2 [ ( − 1 ) n + 1 ( 6 n − 1 ) − 1 ] {\displaystyle 2-8+14-20+26-32+\cdots +(-1)^{n+1}(6n-4)={\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]}
L : ( − 1 ) n + 1 ( 6 n − 4 ) = ( − 1 ) 2 ( 6 − 4 ) = 2 R : 1 2 [ ( − 1 ) n + 1 ( 6 n − 1 ) − 1 ] = 1 2 [ ( − 1 ) 2 ( 6 − 1 ) − 1 ] = 1 2 ∗ 4 = 2 2 = 2 {\displaystyle {\begin{aligned}&L:(-1)^{n+1}(6n-4)=(-1)^{2}(6-4)=2\\&R:{\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]={\frac {1}{2}}[(-1)^{2}(6-1)-1]={\frac {1}{2}}*4=2\\&2=2\\\end{aligned}}}
2 − 8 + 14 − 20 + 26 − 32 + ⋯ + ( − 1 ) k + 1 ( 6 k − 4 ) = 1 2 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] {\displaystyle 2-8+14-20+26-32+\cdots +(-1)^{k+1}(6k-4)={\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]}
2 − 8 + 14 − 20 + 26 − 32 + ⋯ + ( − 1 ) k + 1 ( 6 k − 4 ) ⏟ = 1 2 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] + ( − 1 ) k + 2 ∗ [ 6 ( k + 1 ) − 4 ] = 1 2 [ ( − 1 ) k + 2 ∗ ( 6 k + 5 ) − 1 ] 1 2 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] + ( − 1 ) k + 2 ∗ [ 6 ( k + 1 ) − 4 ] = 1 2 [ ( − 1 ) k + 2 ∗ ( 6 k + 5 ) − 1 ] 1 2 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] + ( − 1 ) k + 2 ∗ ( 6 k + 2 ) = 1 2 [ ( − 1 ) k + 2 ∗ ( 6 k + 5 ) − 1 ] ( − 1 ) k + 1 ( 6 k − 1 ) − 1 + 2 ( − 1 ) k + 2 ∗ ( 6 k + 2 ) = ( − 1 ) k + 2 ∗ ( 6 k + 5 ) − 1 − ( − 1 ) k + 2 ( 6 k − 1 ) + 2 ( − 1 ) k + 2 ∗ ( 6 k + 2 ) = ( − 1 ) k + 2 ∗ ( 6 k + 5 ) − ( 6 k − 1 ) + 2 ( 6 k + 2 ) = ( 6 k + 5 ) − 6 k + 1 + 12 k + 4 = 6 k + 5 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {2-8+14-20+26-32+\cdots +(-1)^{k+1}(6k-4)} _{={\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]}+(-1)^{k+2}*[6(k+1)-4]={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&{\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}*[6(k+1)-4]={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&{\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}*(6k+2)={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&(-1)^{k+1}(6k-1)-1+2(-1)^{k+2}*(6k+2)=(-1)^{k+2}*(6k+5)-1\\&-(-1)^{k+2}(6k-1)+2(-1)^{k+2}*(6k+2)=(-1)^{k+2}*(6k+5)\\&-(6k-1)+2(6k+2)=(6k+5)\\&-6k+1+12k+4=6k+5\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
![{\displaystyle 2-8+14-20+26-32+\cdots ={\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/053964b6074b8ceae18d7be0a11dc3d09e2eb0a2)
![{\displaystyle 2-8+14-20+26-32+\cdots +(-1)^{n+1}(6n-4)={\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d94e1fb8b95cc69be0d9f41a36ebf03aa9397ee5)
![{\displaystyle {\begin{aligned}&L:(-1)^{n+1}(6n-4)=(-1)^{2}(6-4)=2\\&R:{\frac {1}{2}}[(-1)^{n+1}(6n-1)-1]={\frac {1}{2}}[(-1)^{2}(6-1)-1]={\frac {1}{2}}*4=2\\&2=2\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1402c5eb2d4f4d8a4f3155d17bb35a59a7f8392)
טבעי![{\displaystyle 2-8+14-20+26-32+\cdots +(-1)^{k+1}(6k-4)={\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59093e207709af1107b71bfe3a82097777cc6631)
![{\displaystyle {\begin{aligned}&\underbrace {2-8+14-20+26-32+\cdots +(-1)^{k+1}(6k-4)} _{={\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]}+(-1)^{k+2}*[6(k+1)-4]={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&{\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}*[6(k+1)-4]={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&{\frac {1}{2}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}*(6k+2)={\frac {1}{2}}[(-1)^{k+2}*(6k+5)-1]\\&(-1)^{k+1}(6k-1)-1+2(-1)^{k+2}*(6k+2)=(-1)^{k+2}*(6k+5)-1\\&-(-1)^{k+2}(6k-1)+2(-1)^{k+2}*(6k+2)=(-1)^{k+2}*(6k+5)\\&-(6k-1)+2(6k+2)=(6k+5)\\&-6k+1+12k+4=6k+5\\&0=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5978e330f41363ac05a54b8f8bc7516cd7a993a2)
