已知数列 an 的前n项和为sn,满足an+Sn=3-8/2的n次方设bn=2的n次方乘an求证;数列bn是等差数列
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/21 03:24:24
![已知数列 an 的前n项和为sn,满足an+Sn=3-8/2的n次方设bn=2的n次方乘an求证;数列bn是等差数列](/uploads/image/z/1240670-38-0.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97+an+%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAsn%2C%E6%BB%A1%E8%B6%B3an%2BSn%3D3-8%2F2%E7%9A%84n%E6%AC%A1%E6%96%B9%E8%AE%BEbn%3D2%E7%9A%84n%E6%AC%A1%E6%96%B9%E4%B9%98an%E6%B1%82%E8%AF%81%3B%E6%95%B0%E5%88%97bn%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97)
xŒJ@_eC[I_"`_@E"ՄzhsUh0ݤ݄IadUAKIBP<$kX9G˯PI$|X
7>x$4@+ gcdz~"L
4b"8U-7b6[:V/:H:3o{g WK=\yrz
dM;ǵ
T$mZs$aG.}wJA"Ȱ,"GRـ=[Yҽ1R7aܧ!_ɕt3~ޡtL 3zZ
已知数列 an 的前n项和为sn,满足an+Sn=3-8/2的n次方设bn=2的n次方乘an求证;数列bn是等差数列
已知数列 an 的前n项和为sn,满足an+Sn=3-8/2的n次方设bn=2的n次方乘an
求证;数列bn是等差数列
已知数列 an 的前n项和为sn,满足an+Sn=3-8/2的n次方设bn=2的n次方乘an求证;数列bn是等差数列
由题:
Sn = 3 - 8/2^n - an
Sn-1 = 3 - 8/2^(n-1) - an-1
an = Sn - Sn-1 = [3 - 8/2^n - an] - [3 - 8/2^(n-1) - an-1]
= 8/2^(n-1) - 8/2^n - an + an-1
两边同时 +an:
2an = 8/2^(n-1) - 8/2^n + an-1
两边同乘以2^(n-1)
2^n*an = 8 - 8/2 + 2^(n-1)an-1 = 4 + 2^(n-1)*an-1 ——(*)
已知:
bn = 2^n*an
bn-1 = 2^(n-1)*an-1
代入(*)式得:bn = 4 + bn-1
因此bn是等差数列.