问题 解答题
已知数列an的前n项和为Sn,a1=2,nan+1=Sn+n(n+1),
(1)求数列an的通项公式;
(2)设bn=
Sn
2n
,如果对一切正整数n都有bn≤t,求t的最小值.
答案

(1)∵nan+1=Sn+n(n+1)

∴(n-1)an=Sn-1+n(n-1)(n≥2)

两式相减可得,nan+1-(n-1)an=Sn-Sn-1+2n

即nan+1-(n-1)an=an+2n,(n≥2)

整理可得,an+1=an+2(n≥2)(*)

由a1=2,可得a2=S1+2=4,a2-a1=2适合(*)

故数列{an}是以2为首项,以2为公差的等差数列,由等差数列的通项公式可得,an=2+(n-1)×2=2n

(2)由(1)可得,Sn=n(n+1),

bn=

Sn
2n
=
n(n+1)
2n

由数列的单调性可知,bk≥bk+1,bk≥bk-1

k(k+1)
2k
(k+2)(k+1)
2k+1
k(k+1)
2k
k(k-1)
2k-1
解不等式可得2≤k≤3,k∈N*,k=2,或k=3,

b2=b3=

3
2
为数列{bn}的最大项

由bn≤t恒成立可得t≥

3
2
,则t的最小值
3
2

判断题
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Recently some American scientists have given a useful piece of advice to people in industrialized nations. They say people should eat more of the same kind of food eaten by humans living more than 10,000 years ago. ___小题1:___

The scientists say that the human life has changed greatly. Our bodies have not been able to deal with these changes in lifestyle and this had led to new kinds of sickness. ___小题2:___ So they are called “disease of civilization”. Many cancers and diseases of the blood system are examples of such diseases.

Scientists noticed that people in both the Old Stone Age and the New Stone Age enjoyed very little alcohol or tobacco, probably none. __小题3:___ However, a change in food is one of the main differences between life in ancient times and that of day.

Stone Age people hunted wild animals for their meat, which had much less fat than domestic ones. They ate a lot of fresh wild vegetables and fruits. They did not have milk or any other dairy product, and they made very little use of grains.

___小题4:___ We eat six times more salt than our ancestors(祖先). We eat more sugar. We eat twice as much fat but only one third as much protein and much less vitamin C.

___小题5:___ But scientists say that we would be much healthier if we eat much the same way as our ancestors did, cutting the amount of fatty, salty and sweet food.

A.These new sicknesses were not known in ancient times.

B.People today probably don’t want to live the way people thousands of years ago did.

C.Ancient people also lived in large groups.

D.But today, we eat a lot of these.E. Ancient people also got a great deal of physical exercise.

F. In that case, they would live much healthier.

G. People today probably live the same life as people thousands of years ago did.