Question 574789: The sum of the first three terms of a geometric series is 14 and the product of the first three terms of this series is 64.
1. So determine the ratio?
2.Show that there are two different series that satisfy the above conditions.
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Suppose the first three terms are a, ar, ar^2. Then we have
 and 
Since ar = 4, we can replace this into the first equation to get:
 . I'll skip the algebra, but you can show that this is equivalent to
If this quadratic has two real roots r, then we know there are two different common ratios. We can even prove this intuitively; if there are three terms {a,b,c} in a series satisfying the given conditions, then {c,b,a} must also satisfy, and the common ratio is the reciprocal of the original one.
Solving for r, we have
Since ar = 4, we can replace our known values of r to obtain a = 2 or a = 8. We have either {2,4,8} or {8,4,2}.
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