Please help! Advanced Algebra 2!

Question

1.) Three numbers form a geometric sequence whose common ratio is 0.5. If the first is reduced to 10 more than one quarter its value, the second decreased by 10, and the third increased by 10 more than twice its value, the resulting three numbers form an arithmetic sequence. Determine the original three numbers.

I have been stuck on this problem for too long. Help would be greatly appreciated, thank you!

Answer 1

Let's call these three numbers x₁, x_2, and x_3. Since it is a geometric sequence with ratio we have:

x₂ = .5*x₁ and x₃ = .5*x₂

You've said that the following numbers make an arithmetic sequence:

1. .25*x₁ + 10
2. x₂ - 10
3. 2*x₃ + 10

That means that:

(x₂ - 10) - (.25*x₁ + 10) = (2*x₃ + 10) - (x₂ - 10)

which simplifies to:

(-2*x₃ + 2*x₂ - .25*x₁) = 40

Now we substitute in x₁ for x₂ and x₃ using the equations above:

(-.5*x₁ + x₁ - .25x₁) = 40

.25*x₁ = 40

x₁ = 160, x₂ = 80, x₃ = 40