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Four From Arithmetic and Geometric (Posted on 2016-04-29) Difficulty: 3 of 5
Each of X, Y and Z is a distinct positive integer such that
X, Y and Z are in arithmetic sequence, and
X, Y and Z+2016 are in geometric sequence.

Find the four smallest values of X+Y+Z.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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All solutions (spoiler) | Comment 1 of 3
From the arithmetic: z=2y-x
From the geometric: z=y^2/x-2016
Setting the RHS of each equal and setting to zero gives
y^2/x-2y+x-2016=0
Which is a quadratic in y.
Solving for y gives y=x(+/-)x*sqrt(2016/x)
The (-) gives negative solutions
The (+) is easily searchable and the solutions are
(x,y,z)
(14,182,350)
(56,392,728)
(126,630,1124)
(224,896,1568)
(504,1512,2520)
(2016,4032,6048)
Since X+Y+Z=3Y these are in order.
The discriminant is asymptotic to zero and the last solution has discriminant 1 so this is the complete list.

On reflection, this could have been done analytically.

Edited on April 29, 2016, 1:16 pm
  Posted by Jer on 2016-04-29 12:15:45

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