设数列{bn}的前n项和为sn,且bn=2-2sn;数列{an}为等差数列,且a5=14,a7=20求若Cn=An*Bn（n=1,2,3…）,Tn为数列｛Cn｝的前n项和,求证：Tn

来源：学生作业帮助网编辑：作业帮时间：2024/05/06 11:29:17

设数列{bn}的前n项和为sn,且bn=2-2sn;数列{an}为等差数列,且a5=14,a7=20求若Cn=An*Bn（n=1,2,3…）,Tn为数列｛Cn｝的前n项和,求证：Tn
设数列{bn}的前n项和为sn,且bn=2-2sn;数列{an}为等差数列,且a5=14,a7=20求
若Cn=An*Bn（n=1,2,3…）,Tn为数列｛Cn｝的前n项和,求证：Tn

设数列{bn}的前n项和为sn,且bn=2-2sn;数列{an}为等差数列,且a5=14,a7=20求若Cn=An*Bn（n=1,2,3…）,Tn为数列｛Cn｝的前n项和,求证：Tn
b1 = 2-2*b1
3b1 = 2
b1 = 2/3
bn - bn-1 = (2-2sn) - (2-2sn-1) = -2(sn- sn-1) = -2bn
3bn = bn-1
bn = 1/3 * bn-1
{bn}是等比数列
{bn} = { 2/3 * (1/3)^(n-1) } = { 2 * (1/3)^n }
a7 - a5 = 2d
d = (20 - 14 )/2 = 3
a1 = a5 - 4d = 14 - 4*3 = 2
{an} = { 2+ 3(n-1)} = { 3n -1 }
cn = an * bn = (2* (1/3)^n )*(3n-1)) = (6n-2) (1/3)^n
Tn = c1 + c2 +c3 + ...+ cn
Tn - 1/3 Tn =( c1 + c2 +c3 + ...+ cn) - 1/3(c1 + c2 +c3 + ...+ cn)
= c1 + (c2 - 1/3 c1) + (c3- 1/3 c2) + ...+ (cn -1/3 cn-1) - cn
= 4/3 + 6 * (1/3)^2 + 6 *(1/3)^3 + ...+ 6 * (1/3)^n - (6n-2)*(1/3)^n
= 4/3 + 6*( (1/3)^2 - (1/3)^(n+1)) / (1-1/3) ) -(6n-2)*(1/3)^n
= 4/3 + ( 1- (1/3)^(n-1) ) - (6n-2)*(1/3)^n
= 7/3 - (1/3)^(n-1) - (6n-2)*(1/3)^n
= 7/3 - 3*(1/3)^n -6n *(1/3)^n + 2 *(1/3)^n
= 7/3 - (1/3)^n - 6n *(1/3)^n
< 7/3
所以 2/3 *Tn < 7/3
Tn < 7/3 * 3/2 = 7/2

全部展开

设数列{an}为等差数列,且a5=14,a7=20。又等比数列{bn}的前n项和Sn，有S1+S2+S3=22，且a1=b1
(1)求数列 {an}的通项公式。
（2）求数列{bn}的通项公式
a5=a1+4d=14,a7=a1+6d=20，
所以d=3，a1=2，
所以 {an}=3n-1
a1=b1=2，
S1+S2+S3=b1+b1+b1*q+b1+b1*q+b1*q方=2+2+2q+2+2q+2*q方=22
解得q=2或q=-4
{bn}=2 ^n，或分段{bn}=2*（-4） ^（n-1） (n≥2)
=2 (n=1)

收起