在公差不为0的等差数列{an}与递增的等比数列{bn}中,a1=b1=1,a3=b3,a7=b5,则数列{an}中与b9相等的是第几项

Question

在公差不为0的等差数列{an}与递增的等比数列{bn}中,a1=b1=1,a3=b3,a7=b5,则数列{an}中与b9相等的是第几项

Answer 1

a(n) = 1 + (n-1)d,
b(n) = r^(n-1),r > 0.n = 1,2,...
a(3) = 1 + 2d = b(3) = r^2,
a(7) = 1 + 6d = b(5) = r^4 = (1+2d)^2 = 1 + 4d + 4d^2,
0 = 4d^2 - 2d = 4d(d - 1/2),d = 1/2,
r^2 = 1 + 2d = 2,r = 2^(1/2).
b(9) = [2^(1/2)]^8 = 2^4 = 16 = 1 + (n-1)/2,
32 = 2 + n-1,n = 31.
数列{an}中与b9相等的是第31项