SOLUTION: three numbers are in G.P. Their sum is 43 and their product is 216. Find the numbers in the proper order

Question 1122505: three numbers are in G.P. Their sum is 43 and their product is 216. Find the numbers in the proper order
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

Let the numbers be a, ar, and ar^2. Then

(1)
(2)

From (2), we can see immediately that the middle term ar is 6, because (ar)^3 = 216.

There are many ways to find the solution from there. Perhaps one of the easiest ways is to note that the numbers almost certainly have to be whole numbers, in order for both the sum and product to be whole numbers. So if the middle term is 6, the possible common ratios are 2, 3, and 6, giving us the following sets of 3 numbers:

3,6,12
2,6,18
1,6,36

The last one has the required sum of 43, so it is the answer.

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