Question 743687: 3 numbers add to 27 and multiply together to make 288. These numbers also fall into an arithmetic sequence. Show your work.
With the use of guess and check, I have found the answers to be 2, 9, and 16. They add to 27, multiply to 288, and are in an arithmetic sequence. I don't know how to solve this algebraically, however, especially within the series and sequence unit. Any help would be apreciated, thanks.
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
3 numbers add to 27
x + y + z = 27
and multiply together to make 288.
xyz = 288
These numbers also fall into an arithmetic sequence.
That just means that they are equally spaced. That is,
the diffence between the 2nd and 1st equals the difference
between the 3rd and 2nd.
y - x = z - y
which simplifies to
-x + 2y - z = 0
So we have the system:
x + y + z = 27
xyz = 288
-x + 2y - z = 0
Adding the 1st and 3rd equations term by term
x + y + z = 27
-x + 2y - z = 0
----------------
3y = 27
y = 9
Substitute that in the first and 2nd equations:
x + 9 + z = 27 x(9)z = 288
x + z = 18 xz = 32
So now we have the system
x + z = 18
xz = 32
Solve the first for z: z = 18 - x
Substitute in the second equation:
x(18 - x) = 32
18x - x² = 32
-x² + 18x - 32 = 0
x² - 18x + 32 = 0
(x - 16)(x - 2) = 0
x - 16 = 0; x - 2 = 0
x = 16; x = 2
Using x = 16 Using x = 2
z = 18 - x z = 18 - x
z = 18 - 16 z = 18 - 2
z = 2 z = 16
So x and z can switch places. One is 16 and the other is 2
So the arithmetic sequence can either go:
2, 9, 16 where the common difference is 7
or
16, 9, 2 where the common difference is -7
Edwin
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