用数学归纳法证明:(1/1*4)+(1/4*7)+(1/7*10)+.+1/(3n-2)(3n+1)=n/3n+1,(n属於N+)

Question

用数学归纳法证明:(1/1*4)+(1/4*7)+(1/7*10)+.+1/(3n-2)(3n+1)=n/3n+1,(n属於N+)

Answer 1

证明:(1)当n=1 时,左边=1/ 1*4=1/4 =1(3*1+1)命题成立.
（2）假设n=k（k≥1）时,命题成立.
即1/ 1*4+1/4*7+.1/(3k-2)(3k+1)=k/(3k+1);
n=k+1时 ,
则1/ 1*4+1/4*7+.+1/(3k-2)(3k+1)+1)+1/(3k+1)(3k+4)
=k/(3k+1)+1/(3k+1)(3k+4)
=(3k²+4k+1)/(3k+1)(3k+4)
=(3k+1)(k+1)/(3k+1)(3k+4)
=(k+1)/(3k+4)
=(k+1)/[3(k+1)+1]
所以,对于n属於N+命题(1/1*4)+(1/4*7)+(1/7*10)+.+1/(3n-2)(3n+1)=n/3n+1成立.