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)
+23252 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)
a2 + 4ad + 4d2 = a2 + 10ad
4ad + 4d2 = 10ad
4d2 = 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
