设数列{an}的前n项和为sn,且a1=1,sn=nan-2n(n-1)(n∈正整数)证明,证明1/a1a2+1/a2a3+.+1/an-an+1

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设数列{an}的前n项和为sn,且a1=1,sn=nan-2n(n-1)(n∈正整数)证明,证明1/a1a2+1/a2a3+.+1/an-an+1

设数列{an}的前n项和为sn,且a1=1,sn=nan-2n(n-1)(n∈正整数)证明,证明1/a1a2+1/a2a3+.+1/an-an+1
设数列{an}的前n项和为sn,且a1=1,sn=nan-2n(n-1)(n∈正整数)证明,证明1/a1a2+1/a2a3+.+1/an-an+1

设数列{an}的前n项和为sn,且a1=1,sn=nan-2n(n-1)(n∈正整数)证明,证明1/a1a2+1/a2a3+.+1/an-an+1
Sn=nan-2n(n-1)
Sn=n(Sn-S(n-1))-2n(n-1)
(n-1)Sn -nS(n-1) = 2n(n-1)
Sn/n -S(n-1)/(n-1) = 2
Sn/n -S1/1 = 2(n-1)
Sn/n = 2n-1
Sn = n(2n-1)
an = Sn-S(n-1)
= n(2n-1) -(n-1)(2n-3)
=2n^2-n -( 2n^2-5n+3)
= 4n-3
1/[an.a(n+1)] = 1/[(4n-3)(4n-7)]
= (1/4)[ 1/(4n-7)-1/(4n-3) ]
1/(a2a3)+1/(a3a4)+.+1/(an.a(n+1))
=(1/4)[ 1 - 1/(4n-3) ]
< 1/4

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