Coffee Time Challenge
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 = x2.
We know difference of adjacent pairs in the arithmetic progression are constant, so substituting in the result from above we get (6 – x) = (x2 - 6).
We can rearrange this to x2 + 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.