Question

if 2,x,y,-250 is a geometric progression find x and y

Answer 1

In a geometric progression (GP), the ratio of consecutive terms is constant. Let's denote the terms of the GP as follows:

- The first term \( a = 2 \)
- The second term \( b = x \)
- The third term \( c = y \)
- The fourth term \( d = -250 \)

Since this is a GP, we have the following ratios:

\[
\frac{x}{2} = \frac{y}{x} = \frac{-250}{y}
\]

Let's denote the common ratio as \( r \). Therefore, we can express \( x \), \( y \), and \( -250 \) in terms of \( r \):

1. \( x = 2r \)
2. \( y = xr = 2r^2 \)
3. \( -250 = yr = 2r^3 \)

Now we have three equations:

1. \( x = 2r \)
2. \( y = 2r^2 \)
3. \( -250 = 2r^3 \)

From the third equation, we can solve for \( r \):

\[
r^3 = \frac{-250}{2} = -125
\]
\[
r = \sqrt[3]{-125} = -5
\]

Now that we have \( r = -5 \), we can find \( x \) and \( y \):

1. **Finding \( x \)**:
\[
x = 2r = 2 \times -5 = -10
\]

2. **Finding \( y \)**:
\[
y = 2r^2 = 2 \times (-5)^2 = 2 \times 25 = 50
\]

Thus, the values of \( x \) and \( y \) are:

\[
\boxed{-10} \quad \text{and} \quad \boxed{50}
\]