Question

数学归纳法证明1/n+1+1/n+2+1/n+3+...+1/3n>9/10 n>=2

1）当n=2时，左=1/3 +1/4+1/5+1/6=57/60>54/60=9/10，成立．
（2）假设n=k时，有1/(k+1) +1/(k+2) +...+1/3k >9/10
那么　1/(k+2)+1/(k+3) +...+1/3(k+1)
=[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)
>9/10 +1/(3k+3) +1/(3k+3)+1/(3k+3) -1/(k+1)
=9/10
即n=k+1时命题也成立，
从而　原不等式对n∈N，且n>1成立．

第二步中为什么是
>9/10 +1/(3k+3) +1/(3k+3)+1/(3k+3) -1/(k+1)
不应该是
>9/10 +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)的么

Answer 1

1）当n=2时，左=1/3 +1/4+1/5+1/6=57/60>54/60=9/10，成立．
（2）假设n=k时，有1/(k+1) +1/(k+2) +...+1/3k >9/10
那么　1/(k+2)+1/(k+3) +...+1/3(k+1)
=[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)
>9/10 +1/(3k+3) +1/(3k+3)+1/(3k+3) -1/(k+1)

~~~~~~~~~~~~~~~~~~~~~~~~~目的是为了简化计算，这是一道放缩法的应用，在数学中适当的放缩有利于计算~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

=9/10
即n=k+1时命题也成立，
从而　原不等式对n∈N，且n>1成立．

追问

1/(3k+3) +1/(3k+3)+1/(3k+3)=1/(3k+1) +1/(3k+2)+1/(3k+3) ??

追答

当然不等于啦~~~~~~~~~~~~~放说法你们没学过吗？？
1/(3k+3) +1/(3k+3)+1/(3k+3)[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+3) +1/(3k+3)+1/(3k+3)-1/(k+1)，
而[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+3) +1/(3k+3)+1/(3k+3)-1/(k+1)的值是9/10
从而[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)>9/10

~~~~~~
算了做给你看吧~~~~~~~~~~~~~~~~

1）n=2,时，1/3+1/41/5+1/6=19/20>9/10
2）假设n=k时，1/k+1+1/k+2+1/k+3+...+1/3k-1>9/10-1/3k
那么当n=k+1时，
1/k+2+1/k+3+...+1/3k-1+1/3k+1/(3k+1)+1/(3k+2)>9/10+1/3k+1/(3k+1)+1/(3k+2)-1/k+1
那么只需要证明1/3k+1/(3k+1)+1/(3k+2)-1/k+1>-1/(3k+3)
即 1/3k+1/(3k+1)+1/(3k+2）>2/(3k+3)
上式显然成立，那么当n=k+1时，假设也成立
综合1），2）可知道不等式1/n+1+1/n+2+1/n+3+...+1/3n>9/10对于任意n>=2都成立。

Answer 2

写成这样是为了化简9/10 +1/(3k+3) +1/(3k+3)+1/(3k+3) -1/(k+1)=9/10+3/(3k+3)-1/(1+k)=9/10-1/(k+1)-1/(k+1)=9/10

追问

1/(3k+3) +1/(3k+3)+1/(3k+3)=1/(3k+1) +1/(3k+2)+1/(3k+3) ??

追答

不等于，是小于
[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)要大于[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+3) +1/(3k+3)+1/(3k+3)-1/(k+1)，而[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+3) +1/(3k+3)+1/(3k+3)-1/(k+1)的值是9/10所以[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)要大于9/10

追问

我的意思是为什么会突然出现3个1/(3k+1)，其他都懂就这不懂。

追答

是为了好化简[1/(k+1) +1/(k+2)+...+1/3k] +1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)啊，让他消去1/(3k+1) +1/(3k+2)+1/(3k+3) -1/(k+1)后得到一个数，这样好比较啊，如果最后化简得式子中有k，不方便比较