Question

Sn为数列{an}的前N项和,a1=2/9且an=Sn·Sn-1(n>=2)

Sn为数列{an}的前N项和,a1=2/9且an=Sn·Sn-1(n>=2)
(1)求an的通项公式；
(2)n为何值时，|Sn|最大？并求其最大值。

Answer 1

由题意得：
Sn-S(n-1)=an;
1/Sn-1/S(n-1)=(S(n-1) - Sn ) / (Sn·S(n-1)) =-an/an=-1;
S1=a1=2/9;
所以数列{1/Sn}是公差为-1，首项为9/2的等差数列
所以1/Sn=1/S1+(n-1)*(-1)=11/2-n;
所以Sn=2/(11-2n);
所以an=Sn-Sn-1=4/[(2n-11)(2n-13)];

|Sn|=2/|11-2n|
要求|Sn|最大，就要求|11-2n|最小,可得当n=5或6时,|11-2n|最小是1.
即最大值是S5或S6=2

Answer 2

(1)由题意得：an=Sn-Sn-1=Sn*Sn-1
所以：Sn-Sn-1/Sn*Sn-1=1
所以:1/Sn-1/Sn-1=-1
所以：{1/Sn}是以2/9为首项，以-1为公差的等差数列。
所以：1/Sn=(11-2n)/2
所以：Sn=2/(11-2n)
所以：an=Sn-Sn-1=2/(11-2n)-2/(13-2n)=4/(11-2n)(13-2n)

(2)当(11-2n)最小时，Sn最大。
n=5或n=6时，Sn(max)=2

Answer 3

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Answer 5

an=Sn-Sn-1=Sn*Sn-1
Sn(1-Sn-1)=Sn-1
Sn=Sn-1/(1-Sn-1)
1/Sn=(1-Sn-1)/Sn-1=1/Sn-1-1
1/Sn-1/Sn-1=-1，为定值。
{1/Sn}是等差数列。首项=1/a1=9/2，公差=-1。

1/Sn=9/2-(n-1)*1=(11-2n)/2
Sn=2/(11-2n)

an=Sn-Sn-1=2/(11-2n)-2/(13-2n)=4/(11-2n)(13-2n)

当|11-2n|最小时，|Sn|最大。
n=5或n=6时，|Sn|max=2