7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列.

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/06 06:57:34
7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列.
xœN@_#tmVv$y+rERh! @ *B ]"9:`W.ݝfֳ-?/ĭav?2 XV}XSFܹa28.ˈ Aw.qWZvfE޷bqdpyal.W%|>IYy Km(f݂WQdFk0Vڈ${ H1ƃ =C;~BEfA^.0]r^ƹ$z۔||%_"/a3y* PRP|(h\

7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列.
7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列.

7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列.
bn=a1+2a2+3a3+…nan/1+2+3…+n
b(n+1)=[a1+2a2+3a3+…nan+(n+1)a(n+1)]/[1+2+3…+n+(n+1)]
[n(n+1)/2]bn=a1+2a2+3a3+…nan ①
[(n+1)(n+2)/2]b(n+1)=a1+2a2+3a3+…nan+(n+1)a(n+1) ②
②-①得
[(n+1)(n+2)/2]b(n+1)-[n(n+1)/2]bn=(n+1)a(n+1)
两边同时消去(n+1)得
a(n+1)=[(n+2)/2]b(n+1)-(n/2)bn③
an=[(n+1)/2]bn-[(n-1)/2]b(n-1) ④
③-④得a(n+1)-an=[(n+1)/2]b(n+1)+1/2b(n+1)-[(n+1)/2]bn-[(n-1)/2]bn+[(n-1)/2]b(n-1)-1/2bn
=[(n+1)/2][b(n+1)-bn]+1/2[b(n+1)-bn]-[(n-1)/2][bn-b(n-1)]
又{bn}为等差数列,设公差为d
则a(n+1)-an=[(n+1)/2]d+1/2*d-[(n-1)/2]d
=3/2d
所以{an}是公差为3/2d的等差数列
注:此中的an,bn,a(n+1),b(n+1)均是数列中的项
看上去写的有点麻烦,但过程还是比较简单的

设数列{An},{Bn}是公比不相等的两个等比数列,构造新的数列{Cn],满足Cn=An+Bn,求证:数列{Cn}不是等比数列. 设两个数列an,bn 且极限(an-bn)=0 ,n→∞ 数列an,bn 收敛还是发散? 7 设两个数列{an},{bn}满足bn=a1+a2+3a3+…nan/1+2+3…+n,若{bn}为等差数列,求证:{an}也为等差数列. 设数列an,bn是公比不相等的两个等比数列,构造新的数列cn,满足cn=an+bn,求证cn不是等比数列 已知数列{an},{bn}满足a1=2,2an=1+2an*an+1,设{bn}=an-1求数列{1n}为等差数列急!!! 数列 (30 20:12:4)设两个数列{An},{Bn}满足Bn=(a1+2*a2+3*a3+…+n*an)/(1+2+3+…+n),若{Bn}为等差数列,求证:{An}也为等差数列 设数列{An}、{Bn}是公比不相等的两个等比数列,Cn=An+Bn,求证:数列{Cn}不是等比数列. 设数列{An},{Bn}定义如下:. 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,b1=2,a2=3求通项an,bn 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成等比数列,且a1=1,b1=2,a2=3,求通项an,bn 设{an},{bn}是两个数列,点M(1,2),An(2,an),Bn(n-1/n,2/n)为平面直角坐标系内的点.对任意的n属于N*,点点M,An,Bn三点一线,且数列{bn}满足a1b1+a2b2+.+anbn/a1+a2+.+an=2n-3.(1).且数列{an}的通项公式;(2).求证:点p 数列an的前n项和为Sn=2^n-1,设bn满足bn=an+1/an,判断并证明bn 的单调性 设数列an,bn满足:bn=(a1+a2+a3+a4+...+an)/n,若bn是等差数列,求证an也是等差数列 两个数列{an}和{bn}满足bn=a1+2a2+...+nan/1+2+...+n,求证:若{bn}为等差数列,则数列{an}也是等差数列?能看懂的 设数列an前n项和为Sn,且an+Sn=1,求an的通项公式 若数列bn满足b1=1且bn+1=bn+an,求数列bn通项公式 设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn 用数列an表示an+1;并证明;任意n属于设数列{an},{bn}满足;a1=4 a2=5/2,an+1=an+bn/2,bn+1=2anbn/an+bn (1)用数列an表示an+1;并证明;任意n属于N*都 已知各项均为正数的两个数列{an}和{bn}满足:a(n+1)=(an+bn)/√(an²+bn²),n∈N+① 设b(n+1)=1+bn/an,N∈N+,求证数列(bn/an)²是等差数列.②设b(n+1)=(√2)bn/an,且{an}是等比数列,求a1和b1的值.大神给步 数列an=(1/2)^n,数列{bn}满足 bn=3+log4an ,设Tn=|b1|+|b2|+...+|bn|,求Tn .