等比数列{An}的前n项和为Sn,已知对任意的n∈N+点（n,Sn）均在函数y+b^x+r(b>0)且b≠1,b,r均为常数）图像上

题目

等比数列{An}的前n项和为Sn,已知对任意的n∈N+点（n,Sn）均在函数y+b^x+r(b>0)且b≠1,b,r均为常数）图像上
(1)求r的值
（2）当b=2时,记Bn=（n+1）/4An(n∈N+)求数列{Bn}的前n项的Tn.

答案

由题意可知,Sn=b^n+r
所以
An = Sn - S = b^n - b^(n-1)
A = b^(n-1) - b^(n-2)
An/A = b
所以An数列的公比为 b
则
Sn = A1 * (b^n -1)/(b-1) = [A1/(b-1)]*b^n - [A1/(b-1)]
同时
Sn = b^n + r
若对任意n, 以上2式子同时成立, 则
A1/(b-1) = 1
r = -1
---------------------------
当 b = 2 时
A1 = 1
An = A1 * b^(n-1) = 2^(n-1)
Sn = A1 * (b^n -1)/(b-1) = 2^n -1
Bn = n + 1/(4An)
= n + 1/2^(n+1)
Tn = B1 + B2 + …… + Bn
= (1 + 2 + …… + n) + [1/2^2 + 1/2^3 + …… + 1/2^(n+1)]
= n(n+1)/2 + (1/4)*[(1/2)^n - 1]/(1/2 - 1)
= n(n+1)/2 - (1/2)*[(1/2)^n -1]