数列{an}中,a1=1/2,a的n+1项=3an/(an+3),a1·a2+a2·a3+a3·a4+...an·a的n+1项=?

数列{an}中,a1=1/2,a的n+1项=3an/(an+3),a1·a2+a2·a3+a3·a4+...an·a的n+1项=?
数列{an}中,
a1=1/2,a的n+1项=3an/(an+3),
a1·a2+a2·a3+a3·a4+...an·a的n+1项=?

数列{an}中,a1=1/2,a的n+1项=3an/(an+3),a1·a2+a2·a3+a3·a4+...an·a的n+1项=?
两边取倒数
1/a(n+1)=1/a(n)+1/3
{1/a(n)}是一个等差数列
1/a(n)=1/a(1)+(n-1)*1/3
解得a(n)=3/(n+5)
a(n)*a(n+1)
=9/(n+5)(n+6)
=9*(1/(n+5)-1/(n+6))
所以s=9(1/6-1/7+1/7-1/8+1/8-.+1/(n+5)-1/(n+6))
=9(1/6-1/(n+6))
=9n/[6*(n+6)]
=3n/[2*(n+6)]

解:∵a(n+1)=3an/(an+3)
∴a(n+1)*an+3a(n+1)=3an
a(n+1)*an=3[an-a(n+1)]
∴1=3(1/a(n-1)-1/an)]
∴1/a(n-1)-1/an=1/3
∴数列an是以1为首项,1/3为公差的等差数列
∴1/an=a1+(n-1)d
=1/2+(n-1)/3
=(1/6)+...

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解:∵a(n+1)=3an/(an+3)
∴a(n+1)*an+3a(n+1)=3an
a(n+1)*an=3[an-a(n+1)]
∴1=3(1/a(n-1)-1/an)]
∴1/a(n-1)-1/an=1/3
∴数列an是以1为首项,1/3为公差的等差数列
∴1/an=a1+(n-1)d
=1/2+(n-1)/3
=(1/6)+(n/3)
∴a1·a2+a2·a3+a3·a4+...an·a(n-1)
=3(a2-a1)+3(a3-a2)+...+3(an-a(n-1)
=3a2-3a1+3a3-3a2+3a4-3a3+...+3(an-a(n-1)]
=-3a1+3an
=2+n-(3/2)
=(1/2)+n
=-3a1
=

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