find all possible values of the common ratio of the geometric sequence a, b, 6, ... if a+b=1
We know that b = sqrt(6a) = 1-a
so,
6a = (1-a)^2 = 1 - 2a + a^2
a^2 - 8a + 1 = 0
a = 4 ± √15
the ratio r = √(6/a) = √(6/(4+√15)) = √(6(4-√15))
= √(24 - 6√15) = √(9 - 6√15 + 15)
= √(3-√15)^2 = 3-√15
similarly, r=3+√15 if a = 4-√15
It all works out. Crazy, huh?