已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=f(an)(n∈N*)记Sn=a1a2+a2a3+...+anan+1,求Sn

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已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=f(an)(n∈N*)记Sn=a1a2+a2a3+...+anan+1,求Sn

已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=f(an)(n∈N*)记Sn=a1a2+a2a3+...+anan+1,求Sn
已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=f(an)(n∈N*)
记Sn=a1a2+a2a3+...+anan+1,求Sn

已知函数f(x)=x/(3x+1),数列{an}满足a1=1,an+1=f(an)(n∈N*)记Sn=a1a2+a2a3+...+anan+1,求Sn
由题意
a(n+1)=an/(3an+1)
取倒数得
[1/a(n+1)]=[1/an]+3
故数列{1/an}是公差为3首项为1/a1=1的等差数列
故1/an=1+3(n-1)
=3n-2

sn=1/(3n-2)

Sn=1/(1*4)+1/(4*7)...+1/(3n-2)(3n+1)
=(1/3)[1-(1/4)+(1/4).+(1/(3n-2))-1/(3n+1)]
=(1/3)3n/(3n+1)
=n/(3n+1)