< m a t h > 1 − 4 + 7 − 10 + ⋯ + ( − 1 ) n + 1 ( 3 n − 2 ) = 1 4 [ ( − 1 ) n + 1 ( 6 n − 1 ) − 1 ] {\displaystyle <math>1-4+7-10+\cdots +(-1)^{n+1}(3n-2)={\frac {1}{4}}[(-1)^{n+1}(6n-1)-1]}
L : ( − 1 ) n + 1 ( 3 n − 2 ) = ( − 1 ) 2 ( 3 − 2 ) R : 1 4 [ ( − 1 ) n + 1 ( 6 n − 1 ) − 1 ] = 1 [ 4 ] [ ( − 1 ) 2 ∗ 5 − 1 ] = 4 4 = 1 1 = 1 {\displaystyle {\begin{aligned}&L:(-1)^{n+1}(3n-2)=(-1)^{2}(3-2)\\&R:{\frac {1}{4}}[(-1)^{n+1}(6n-1)-1]={\frac {1}{[}}4][(-1)^{2}*5-1]={\frac {4}{4}}=1\\&1=1\\\end{aligned}}}
< m a t h > 1 − 4 + 7 − 10 + ⋯ + ( − 1 ) k + 1 ( 3 k − 2 ) = 1 4 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] {\displaystyle <math>1-4+7-10+\cdots +(-1)^{k+1}(3k-2)={\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]}
1 − 4 + 7 − 10 + ⋯ + ( − 1 ) k + 1 ( 3 k − 2 ) ⏟ = 1 4 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] + ( − 1 ) k + 2 ( 3 k + 1 ) = 1 4 [ ( − 1 ) k + 2 ( 6 k + 5 ) − 1 ] 1 4 [ ( − 1 ) k + 1 ( 6 k − 1 ) − 1 ] + ( − 1 ) k + 2 ( 3 k + 1 ) = 1 4 [ ( − 1 ) k + 2 ( 6 k + 5 ) − 1 ] ( − 1 ) k + 1 ( 6 k − 1 ) − 1 + 4 ( − 1 ) k + 1 ∗ ( − 1 ) ( 3 k + 1 ) = ( − 1 ) k + 1 ( − 1 ) ( 6 k + 5 ) − 1 / + 1 : ( − 1 ) k + 1 ( 6 k − 1 ) − 4 ( 3 k + 1 ) = − 6 k − 5 6 k − 1 − 12 k − 4 = − 6 k − 5 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {1-4+7-10+\cdots +(-1)^{k+1}(3k-2)} _{={\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]}+(-1)^{k+2}(3k+1)={\frac {1}{4}}[(-1)^{k+2}(6k+5)-1]\\&{\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}(3k+1)={\frac {1}{4}}[(-1)^{k+2}(6k+5)-1]\\&(-1)^{k+1}(6k-1)-1+4(-1)^{k+1}*(-1)(3k+1)=(-1)^{k+1}(-1)(6k+5)-1/+1:(-1)^{k+1}\\&(6k-1)-4(3k+1)=-6k-5\\&6k-1-12k-4=-6k-5\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
![{\displaystyle <math>1-4+7-10+\cdots +(-1)^{n+1}(3n-2)={\frac {1}{4}}[(-1)^{n+1}(6n-1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5a81f87c38ff60cbb5edca001f275daeba24aa)
[עריכה]![{\displaystyle {\begin{aligned}&L:(-1)^{n+1}(3n-2)=(-1)^{2}(3-2)\\&R:{\frac {1}{4}}[(-1)^{n+1}(6n-1)-1]={\frac {1}{[}}4][(-1)^{2}*5-1]={\frac {4}{4}}=1\\&1=1\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c207c17be9c21f708bfac3c17048c3477966e260)
טבעי[עריכה]![{\displaystyle <math>1-4+7-10+\cdots +(-1)^{k+1}(3k-2)={\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00ade68ef57d2f3ccd3d0efb041bc49d3bf989c0)
![{\displaystyle {\begin{aligned}&\underbrace {1-4+7-10+\cdots +(-1)^{k+1}(3k-2)} _{={\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]}+(-1)^{k+2}(3k+1)={\frac {1}{4}}[(-1)^{k+2}(6k+5)-1]\\&{\frac {1}{4}}[(-1)^{k+1}(6k-1)-1]+(-1)^{k+2}(3k+1)={\frac {1}{4}}[(-1)^{k+2}(6k+5)-1]\\&(-1)^{k+1}(6k-1)-1+4(-1)^{k+1}*(-1)(3k+1)=(-1)^{k+1}(-1)(6k+5)-1/+1:(-1)^{k+1}\\&(6k-1)-4(3k+1)=-6k-5\\&6k-1-12k-4=-6k-5\\&0=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3aadd5bbecae5e247474277596cd535bb526b6)
