【答案】
分析:(Ⅰ)根據(jù)題中給出的設(shè)數(shù)列{a
n}的前n項和為S
n便可求出數(shù)列{
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/0.png)
}是公差為1的等差數(shù)列,將a1=4代入便可求出數(shù)列{a
n}的通項公式;
(Ⅱ)先求出數(shù)列bn的通項公式,然后求寫前n項和Bn的表達式,進而求出的B
3n-B
n表達式,然后證明B
3n-B
n為遞增數(shù)列,即當(dāng)n=2時,B
3n-B
n最小,便可求出m的最大值.
(Ⅲ)先將所需證明的不等式化簡為
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/1.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/2.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/3.png)
<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/4.png)
,然后利用函數(shù)的導(dǎo)函數(shù)證明g(x)=ln(x+1)-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/5.png)
為增函數(shù),即可證明當(dāng)n∈N*且n≥2時,T
2n<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/6.png)
.
解答:解:(Ⅰ)由S
n=2a
n-2
n+1,得S
n-1=2a
n-1-2
n(n≥2).
兩式相減,得a
n=2a
n-2a
n-1-2
n,即a
n-2a
n-1=2
n(n≥2).
于是
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/7.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/8.png)
=1,所以數(shù)列{
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/9.png)
}是公差為1的等差數(shù)列.(2分)
又S
1=a
1=2a
1-2
2,,所以a
1=4.
所以
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/10.png)
=2+(n-1)=n+1,故a
n=(n+1)•2
n.(4分)
(注:該問也可用歸納,猜想,數(shù)學(xué)歸納法證明的方法)
(Ⅱ)因為b
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/11.png)
=log
2n2=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/12.png)
,則B
3n-B
n=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/13.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/14.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/15.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/16.png)
.
令f(n)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/17.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/18.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/19.png)
,
則f(n+1)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/20.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/21.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/22.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/23.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/24.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/25.png)
.
所以f(n+1)-f(n)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/26.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/27.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/28.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/29.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/30.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/31.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/32.png)
>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/33.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/34.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/35.png)
=0.
即f(n+1)>f(n),所以數(shù)列{f(n)}為遞增數(shù)列.(7分)
所以當(dāng)n≥2時,f(n)的最小值為f(2)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/36.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/37.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/38.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/39.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/40.png)
.
據(jù)題意,
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/41.png)
<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/42.png)
,即m<19.又m為整數(shù),
故m的最大值為18.(8分)
(Ⅲ)證明:因為c
n=(-1)
n+1•
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/43.png)
,則當(dāng)n≥2時,
T
2n=1-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/44.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/45.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/46.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/47.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/48.png)
=(1+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/49.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/50.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/51.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/52.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/53.png)
)-2(
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/54.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/55.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/56.png)
)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/57.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/58.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/59.png)
.(9分)
下面證
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/60.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/61.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/62.png)
<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/63.png)
.
先證一個不等式,當(dāng)x>0時,ln(x+1)>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/64.png)
.
令g(x)=ln(x+1)-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/65.png)
(x>0),則g′(x)=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/66.png)
-
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/67.png)
=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/68.png)
>0,
∴g(x)在(0,+∞)時單調(diào)遞增,
則g(x)>g(0)=0,即當(dāng)x>0時,ln(x+1)>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/69.png)
,
令x=
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/70.png)
,則ln
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/71.png)
>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/72.png)
⇒ln(n+1)-lnn>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/73.png)
,
∴l(xiāng)n(n+2)-ln(n+1)>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/74.png)
,
ln(n+3)-ln(n-2)>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/75.png)
,
…,
ln(2n)-ln(2n-1)>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/76.png)
以上n個式相加,即有l(wèi)n(2n)-lnn>
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/77.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/78.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/79.png)
∴
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/80.png)
+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/81.png)
+…+
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/82.png)
<ln(2n)-lnn<ln2<
![](http://thumb.zyjl.cn/pic6/res/gzsx/web/STSource/20131024181846324096970/SYS201310241818463240969021_DA/83.png)
從而原不等式得證.(14分)
點評:本題主要考查等差數(shù)列、等比數(shù)列、放縮法等基礎(chǔ)知識,考查綜合運用知識分析問題和解決問題的能力,解題時注意整體思想和轉(zhuǎn)化思想的運用,屬于中檔題.