人共得金( )A.斤?B.斤C.2斤?D.斤解析:选D.由题意可知等差数列{an}中,即,解得d=-,所以a4+a5+a6=(a1+a2+a3)+9d=.故选D.12.首项为-24的等差数列{an},从第10项开始为正数,则公差d的取值范围是________.解析:设等差数列的公差为d,则通项公式an=-24+(n-1)d,由解得<d≤3,即公差的取值范围是.答案:13.在数列{an}中,a1=2,an+1=an+2n+1.(1)求证:数列{an-2n}为等差数列;(2)设数列{bn}满足bn=2log2(an+1-n),求{bn}的通项公式.解:(1)证明:(an+1-2n+1)-(an-2n)=an+1-an-2n=1(与n无关),故数列{an-2n}为等差数列,且公差d=1.(2)由第一问可知,an-2n=(a1-2)+(n-1)d=n-1,故an=2n+n-1,所以bn=2log2(an+1-n)=2n.14.(选做题)若数列{bn}对于n∈N+,都有bn+2-bn=d(d为常数),则称数列{bn}是公差为d的准等差数列.=,}是公差为8的准等差数列.设数列{an}满足:a1=a,对于n∈N+,都有an+an+1=2n.(1)求证:数列{an}为准等差数列;(2)求数列{an}的通项公式.解:(1)证明:因为an+an+1=2n(n∈N+),①所以an+1+an+2=2(n+1),②②-①得an+2-an=2(n∈N+),所以数列{an}是公差为2的准等差数列.(2)因为a1=a,an+an+1=2n(n∈N+),所以a1+a2=2×1,即a2=2-a.因为a1,a3,a5,…是以a为首项,2为公差的等差数列,a2,a4,a6,…是以2-a为首项,2为公差的等差数列,所以当n为偶数时,an=2-a+×2=n-a,当n为奇数时,an=a+×2=n+a-1.所以an=.
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