【答案】
分析:本題考查的知識點是數(shù)學(xué)歸納法及極限的運(yùn)算.
(1)由數(shù)列{a
n}滿足(n-1)a
n+1=(n+1)(a
n-1)且a
2=6,設(shè)b
n=a
n+n(n∈N*).我們不難給出數(shù)列{b
n}的前若干項,并能由此歸納推理出數(shù)列的通項公式,但歸納推理的結(jié)論不一定正確,我們可以用數(shù)學(xué)歸納學(xué)進(jìn)行證明.
(2)由(1)的結(jié)論,結(jié)合數(shù)列求和的裂項法,我們不難對

(

+

+

+…+

)進(jìn)行化簡,進(jìn)而求出

(

+

+

+…+

)的值.
解答:解:(1)n=1時,由(n-1)a
n+1=(n+1)(a
n-1),得a
1=1.
n=2時,a
2=6代入得a
3=15.同理a
4=28,
再代入b
n=a
n+n,有b
1=2,b
2=8,b
3=18,b
4=32,由此猜想b
n=2n
2.
要證b
n=2n
2,只需證a
n=2n
2-n.
①當(dāng)n=1時,a
1=2×1
2-1=1成立.
②假設(shè)當(dāng)n=k時,a
k=2k
2-k成立.
那么當(dāng)n=k+1時,由(k-1)a
k+1=(k+1)(a
k-1),得a
k+1=

(a
k-1)
=

(2k
2-k-1)=

(2k+1)(k-1)=(k+1)(2k+1)=2(k+1)
2-(k+1).
∴當(dāng)n=k+1時,a
n=2n
2-n正確,從而b
n=2n
2.
(2)

(

+

+…+

)
=

(

+

+…+

)
=


[

+

+…+

]
=


[1-

+

-

+…+

-

]
=


[1+

-

-

]
=

.
點評:歸納推理的一般步驟是:(1)通過觀察個別情況發(fā)現(xiàn)某些相同性質(zhì);(2)從已知的相同性質(zhì)中推出一個明確表達(dá)的一般性命題(猜想).但歸納推理的結(jié)論不一定正確,我們要利用數(shù)學(xué)歸納法等方法對歸納的結(jié)論進(jìn)行進(jìn)一步的論證.