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问题 解答题
已知数列{an}满足an=2an-1+2n-1（n∈N*，n≥2），且a4=81
（1）求数列的前三项a1、a2、a3的值；
（2）是否存在一个实数λ，使得数列{
an+λ
2n
}为等差数列？若存在，求出λ的值；若不存在，说明理由；求数列an通项公式．
答案

（1）由an=2an-1+2n-1（n≥2）⇒a4=2a3+24-1=81⇒a3=33

同理可得a2=13，a1=5（3分）

（2）假设存在一个实数λ符合题意，则

an+λ
2n
-
an-1+λ
2n-1
必为与n无关的常数

∵

an+λ
2n
-
an-1+λ
2n-1
=
an-2an-1-λ
2n
=
2n-1-λ
2n
=1-
1+λ
2n
（5分）

要使

an+λ
2n
-
an-1+λ
2n-1
是与n无关的常数，则
1+λ
2n
=0，得λ=-1

故存在一个实数λ=-1，使得数列{

an+λ
2n
}为等差数列（8分）

由（2）知数列{

an+λ
2n
}的公差d=1，∴
an-1
2n
=
a1-1
21
+(n-1)•1=n+1

得an=（n+1）•2n+1（13分）

问答题

The first outline of The Ascent of Man was written in July 1969 and the last foot of film was shot in December 1972. An undertaking as large as this, though wonderfully exhilarating, is not entered lightly. It demands an unflagging intellectual and physical vigour, a total immersion, which I had to be sure that I could sustain with pleasure; for instance, I had to put off researches that I had already begun; and I ought to explain what moved me to do so.
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