Question

The numbers x, y, 8 are in GP, and the numbers x, y, minus 8 are in AP. The values of x, y are

Answer 1

Let's first find the common ratio of the geometric progression (GP). Let's call it r.

Since x, y, and 8 are in GP, we have:

y/x = 8/y

Cross multiplying, we get:

y^2 = 8x

Now let's find the common difference of the arithmetic progression (AP). Let's call it d.

Since x, y, and -8 are in AP, we have:

y - x = -8

We have two equations:

1. y^2 = 8x
2. y - x = -8

Let's solve these equations simultaneously.

From equation 2, we have:

x = y + 8

Substituting this value of x into equation 1, we get:

y^2 = 8(y + 8)

Expanding, we have:

y^2 = 8y + 64

Rearranging, we have:

y^2 - 8y - 64 = 0

Factoring, we have:

(y - 16)(y + 4) = 0

So, y = 16 or y = -4.

If y = 16, then x = 16 + 8 = 24.

If y = -4, then x = -4 + 8 = 4.

Therefore, the possible values of x and y are (24, 16) or (4, -4).