两正数数列{an} {bn}满足:an,bn,a(n+1)成等差数列,bn,a(n+1),b(n+1)成等比数列 a1=

来源：学生作业帮编辑：搜狗做题网作业帮分类：数学作业时间：2024/07/22 05:51:58

两正数数列{an} {bn}满足:an,bn,a(n+1)成等差数列,bn,a(n+1),b(n+1)成等比数列 a1=1 b1=2 a2=3.求{an} {bn}通项公式.

①必要性：
设{an}成等差数列,公差为d,∵{an}成等差数列.
bn＝（a1+2a2+…+nan）/（1+2+3+…n）=[a1(1+2+...+n)+d(1*2+2*3+...+(n-1)*n)/(1+n...+n)=a1+2(n-1)d/3,
从而bn+1－bn=a1+2nd/3－a1－2(n－1)d/3=2d/3为常数.?
故{bn}是等差数列,公差为 2d/3.
②充分性:
设{bn}是等差数列,公差为d′,则bn=(n－1)d′?
∵bn*(1+2+…+n)=a1+2a2+…+nan ①
bn－1*(1+2+…+n－1)=a1+2a2+…+(n－1)an ②
①－②得：nan=n(n+1)bn/2-n(n-1)(bn－1)/2?
∴an=(n+1)bn/2-(n-1)(bn-1)/2=(n+1)[b1+(n-1)d']/2-(n-1)[b1+(n-2)d']/2=b1+3(n-1)d'/2 ,
从而得an+1－an= 3d′/2为常数,故{an}是等差数列.
综上所述,数列{an}成等差数列的充要条件是数列{bn}也是等差数列.

有两个正数数列an,bn,对任意正整数n,有an,bn,an+1成等比数列,bn,an+1,bn+1成等差数列,若a1= 高一等比数列证明题,正数列{an}和{bn}满足,对于任意自然数n,an,bn,a(n+1)成等差数列,bn,a(n+1 问道数学题.正数数列｛an}和{bn}满足：对任意自然数n,an,bn,a(n+1)成等差数列,bn.a(n+1)成等比数列an,bn各项均为正数,对任意n,an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列证数列根号BN成数列{an}、{bn}的每一项都是正数,a1=8,b1=16,且an,bn,a(n+1)成等差,bn,a(n+1),b( 已知正项数列{an},{bn}满足:对任意正整数n,都有an,bn,a(n+1)成等差数列,bn,a(n+1),b（n＋在数列{an}，{bn}中，a1=2，b1=4且an，bn，an+1成等差数列，bn，an+1，bn+1成等比数列（n∈ 在数列{an},{bn}中,a1=2,b1=4,且an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列（n 设各项均为正数的数列{an}和{bn}满足:an,bn,an+1成等差数列,bn,an+1,bn+1等比数列且a1=1, 正数列{an}和{bn}满足对任意自然数n,an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列已知数列an,bn,cn满足[a(n+1)-an][b(n+1)-bn]=cn 已知正项数列｛an｝｛bn｝满足,对任意正整数n,都有an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列