A short little puzzle to solve over your next morning coffee:

1, x, y are three numbers in geometric progression.

x, 6, y are in arithmetic progression.

What are the values of x, y ?

Easy Solution

Without even breaking out math, you probably got the solution x = 3, y = 9

This gives the geometric progression of: 1, 3, 9 and the arithmetic progression as: 3, 6, 9.

There is, however, another solution. Can you find it?

Solution

We know from the geometric progression that x/1 = y/x, which means y = x^{2}.

We know difference of adjacent pairs in the arithmetic progression are constant, so substituting in the result from above we get (6 – x) = (x^{2} - 6).

We can rearrange this to x^{2} + x – 12 = 0

And we can factor this to (x + 4)(x – 3) = 0, giving the solutions x = 3 and x = -4.

We already know the solution when x = 3, so the second solution is x = -4, which results in y = 16.

This gives the geometric progression of: 1, -4, 16 and the arithmetic progression as: -4, 6, 16.