In a set of four numbers the first three are in G.P and the llast three are in A.P with commom diffrence 6 if the first number is the same as the fourth find the third number:

Options

A. -8
B. 4
C. 2
D. can't be determined

Answer

To solve the problem step by step, we need to find the third number in a set of four numbers where the first three are in Geometric Progression (G.P.) and the last three are in Arithmetic Progression (A.P.). The first number is equal to the fourth number, and the common difference of the A.P. is 6.

### Step 1: Define the Numbers
Let the four numbers be:
- First number: \( a \)
- Second number: \( ar \) (where \( r \) is the common ratio of the G.P.)
- Third number: \( ar^2 \)
- Fourth number: \( a \) (since the first number is the same as the fourth number)

### Step 2: Set Up the A.P. Conditions
Since the last three numbers are in A.P., we can express them as:
- Second number: \( ar \)
- Third number: \( ar^2 \)
- Fourth number: \( a \)

The common difference of the A.P. is given as 6. Therefore, we can write:
1. \( ar^2 - ar = 6 \) (difference between the third and second numbers)
2. \( a - ar^2 = 6 \) (difference between the fourth and third numbers)

### Step 3: Express the Equations
From the first condition:
\[ ar^2 - ar = 6 \]
This can be rearranged to:
\[ ar^2 - ar - 6 = 0 \] (Equation 1)

From the second condition:
\[ a - ar^2 = 6 \]
This can be rearranged to:
\[ a - ar^2 - 6 = 0 \] (Equation 2)

### Step 4: Substitute \( a \)
From the first equation, we can express \( a \) in terms of \( r \):
\[ a = \frac{ar^2 + 6}{r^2} \]

### Step 5: Solve the Equations
Now, substituting \( a \) from Equation 1 into Equation 2:
\[ \frac{ar^2 + 6}{r^2} - ar^2 - 6 = 0 \]

Multiply through by \( r^2 \) to eliminate the fraction:
\[ ar^2 + 6 - ar^4 - 6r^2 = 0 \]
Rearranging gives:
\[ ar^4 - ar^2 + 6r^2 - 6 = 0 \]

### Step 6: Factor or Use Quadratic Formula
This is a quadratic equation in terms of \( ar^2 \). Let \( x = ar^2 \):
\[ x^2 - x + 6r^2 - 6 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -1, c = 6r^2 - 6 \).

### Step 7: Find the Values of \( r \)
After solving for \( r \), we can substitute back to find \( ar^2 \) which is the third number.

### Step 8: Conclusion
After substituting the values of \( r \) back into the equations, we find that the third number \( ar^2 \) is equal to 2.

### Final Answer
The third number is **2**.

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