SOLUTION: Three consecutive numbers of a G.p are such that there sum is 26 and there product is 216.Find the numbers

Question 858284: Three consecutive numbers of a G.p are such that there sum is 26 and there product is 216.Find the numbers
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE FAST WAY:
If the product is ,

The product of the three terms could be
with the terms being 2, 6, and 18 (or 18, 6, and 2)
or
the product could be
with the terms being 3, 6, and 12 (or 12, 6, and 3)

while , so the numbers are
, , and .
That gives as a solution quickly, without proving that it is the only solution.

THE EXPECTED WAY (with lots of formulas and calculations):
We will call the first term we are looking for ,
and the common ratio .
The formula for term number is .
The second and third terms would be
and .
Their product would be
We could calculate their sum as

Otherwise, the formula for the sum of the first terms is
.
The sum of the three terms would be

Our equations are
or and
or or .
From we get -->-->
If we substitute either one into we get an equation in one variable that we can solve.
-->-->-->--><-->
No matter how we solve or
we find .
-->-->--> the terms , in order, are 2, 6, and 18.
-->-->-->--> the terms , in order, are 18, 6, and 2.
The terms (regardles of order are , , and ,
and that is the only solution
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