Four numbers are in G.P. The sum of first two numbers is 4 and the sum of last two numbers is 36. Find the numbers.

Explanation

To solve the problem of finding four numbers in a geometric progression (G.P.) given that the sum of the first two numbers is 4 and the sum of the last two numbers is 36, we can follow these steps:

### Step 1: Define the numbers in G.P.
Let the four numbers in G.P. be:
- First number: \( a \)
- Second number: \( ar \)
- Third number: \( ar^2 \)
- Fourth number: \( ar^3 \)

### Step 2: Set up the equations based on the problem statement
From the problem, we have two conditions:
1. The sum of the first two numbers:
\[ a + ar = 4 \]
This can be factored as:
\[ a(1 + r) = 4 \quad \text{(Equation 1)} \]

2. The sum of the last two numbers:
\[ ar^2 + ar^3 = 36 \]
This can be factored as:
\[ ar^2(1 + r) = 36 \quad \text{(Equation 2)} \]

### Step 3: Express \( a \) in terms of \( r \)
From Equation 1, we can express \( a \):
\[ a = \frac{4}{1 + r} \]

### Step 4: Substitute \( a \) into Equation 2
Substituting \( a \) into Equation 2:
\[ \left(\frac{4}{1 + r}\right)r^2(1 + r) = 36 \]
This simplifies to:
\[ 4r^2 = 36 \]

### Step 5: Solve for \( r^2 \)
Dividing both sides by 4:
\[ r^2 = 9 \]
Taking the square root:
\[ r = 3 \quad \text{or} \quad r = -3 \]

### Step 6: Find \( a \) for both values of \( r \)
1. **If \( r = 3 \)**:
\[ a = \frac{4}{1 + 3} = \frac{4}{4} = 1 \]

2. **If \( r = -3 \)**:
\[ a = \frac{4}{1 - 3} = \frac{4}{-2} = -2 \]

### Step 7: Calculate the four numbers for both cases
1. **For \( r = 3 \)**:
- First number: \( a = 1 \)
- Second number: \( ar = 1 \cdot 3 = 3 \)
- Third number: \( ar^2 = 1 \cdot 3^2 = 9 \)
- Fourth number: \( ar^3 = 1 \cdot 3^3 = 27 \)

So, the numbers are \( 1, 3, 9, 27 \).

2. **For \( r = -3 \)**:
- First number: \( a = -2 \)
- Second number: \( ar = -2 \cdot (-3) = 6 \)
- Third number: \( ar^2 = -2 \cdot 9 = -18 \)
- Fourth number: \( ar^3 = -2 \cdot (-27) = 54 \)

So, the numbers are \( -2, 6, -18, 54 \).

### Conclusion
The two sets of four numbers in G.P. that satisfy the given conditions are:
1. \( 1, 3, 9, 27 \)
2. \( -2, 6, -18, 54 \)