The third term of an arithmetic sequence of positive numbers, in which the difference between the terms is not zero, is the geometric mean of the first and eleventh terms. What is the ratio of the second term to the first term of the sequence?

The geometric sequence is basically if you want to find the geo mean of a and b, that's sqrt(ab)

Guest Aug 6, 2020

#1**0 **

First term: a

Second term: a + d

Third term: a + 2d

Eleventh term: a + 10d

Third term is the geometric mean of the first term and the eleventh term:

a + 2d = sqrt( a · (a + 10d) ) ---> (a + 2d)^{2} = a(a + 10d)

a^{2} + 4ad + 4d^{2} = a^{2} + 10ad

4ad + 4d^{2} = 10ad

4d^{2} = 6ad

2d = 3a

d = (3/2)a

Ratio of the second term to the first term:

(a + d) / a ---> ( a + (3/2)a ) / a

(2a + 3a) / (2a) (mult num and den by 2)

5a / 2a

5/2

geno3141 Aug 6, 2020