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当前位置:首页 问答 小学 数学 已知数列满足:a1=1,a(n+1)=an+1,n为奇数;2an,n为偶数,设bn=a2n-1,(Ⅰ)求b2,b3,并证明:b(n+1)=2bn+2;(Ⅱ)①证明:数列{bn+2}为等比数列;②若a2k,a(2k+1),9+a(2k+2)成等比数列,求正整数k的值.
已知数列满足:a1=1,a(n+1)=an+1,n为奇数;2an,n为偶数,设bn=a2n-1,(Ⅰ)求b2,b3,并证明:b(n+1)=2bn+2;(Ⅱ)①证明:数列{bn+2}为等比数列;②若a2k,a(2k+1),9+a(2k+2)成等比数列,求正整数k的值.
更新时间:2025-04-10 21:15:08 专题:数学
问题描述:

已知数列满足:a1=1,a(n+1)=an+1,n为奇数;2an,n为偶数,设bn=a2n-1,

(Ⅰ)求b2,b3,并证明:b(n+1)=2bn+2;

(Ⅱ)①证明:数列{bn+2}为等比数列;

②若a2k,a(2k+1),9+a(2k+2)成等比数列,求正整数k的值.

宋永明回答:   (I)   a(n+1)=an+1,nisodd   =2an,niseven   bn=a(2n)-1   a1=1   a2=a1+1=2   ifnisodd,   a(n+1)=an+1   =2a(n-1)+1   a(n+1)+1=2[(a(n-1)+1]   a(n+1)+1=2^[(n-1)/2].(a2+1)   =3.2^[(n-1)/2]   a(n+1)=-1+3.2^[(n-1)/2]   nisodd=>n=2m-1   a(2m)=-1+3.2^(m-1)   bn=a(2n)-1   =-2+3.2^(n-1)   b2=-2+3.2=4   b3=-2+3.4=10   b(n+1)=-2+3.2^n   =2(-2+3.2^(n-1))+2   =2bn+2   (II)   (1)   bn+2=3.2^(n-1)   {bn+2}是等比数列,q=3   (2)   ifniseven,   a(n+1)=2an   =2(a(n-1)+1)   a(n+1)+2=2(a(n-1)+2)   =2^(n/2).(a1+2)   =3.2^(n/2)   a(n+1)=-2+3.2^(n/2)   niseven,n=2k   a(2k+1)=-2+3.2^k   a(2k),a(2k+1),9+a(2k+2)成等比数列   a(2k).[9+a(2k+2)]=[a(2k+1)]^2   [-1+3.2^(k-1)].(8+3.2^k)=(-2+3.2^k)^2   -8-3.2^k+12.2^k+9.2^(2k-1)=4-12.2^k+9.2^(2k)   (9/2).2^(2k)-21.2^k+12=0   9.2^(2k)-42.2^k+24=0   (2^k-4)(9.2^k-6)=0   2^k=4   k=2
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《已知数列满足:a1=1,a(n+1)=an+1,n为奇数;2an,n为偶数,设bn=a2n-1,(Ⅰ)求b2,b3,并证明:b(n+1)=2bn+2;(Ⅱ)①证明:数列{bn+2}为等比数列;②若a2k,a(2k+1),9+a(2k+2)成等比数列,求正整数k的值.-酷奇网问答》
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