Find three numbers in G.P. such that their sum is 14 and the sum of their squares is 84-

Options

A. 3,6,12
B. 2,6,18
C. 1,3,9
D. 2,4,8

Answer

To find three numbers in Geometric Progression (G.P.) such that their sum is 14 and the sum of their squares is 84, we can follow these steps:

### Step 1: Define the terms in G.P.
Let the three numbers in G.P. be:
- First term: \( \frac{A}{R} \)
- Second term: \( A \)
- Third term: \( AR \)

### Step 2: Set up the equations
According to the problem:
1. The sum of the numbers:
\[ \frac{A}{R} + A + AR = 14 \quad \text{(Equation 1)} \]
2. The sum of their squares:
\[ \left(\frac{A}{R}\right)^2 + A^2 + (AR)^2 = 84 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 1
Multiply through by \( R \) to eliminate the fraction:
\[ A + AR + A R^2 = 14R \]
Rearranging gives:
\[ A(1 + R + R^2) = 14R \quad \text{(Equation 3)} \]

### Step 4: Simplify Equation 2
Expanding the squares gives:
\[ \frac{A^2}{R^2} + A^2 + A^2R^2 = 84 \]
Multiplying through by \( R^2 \) to eliminate the fraction:
\[ A^2 + A^2R^2 + A^2R^4 = 84R^2 \]
Factoring out \( A^2 \):
\[ A^2(1 + R^2 + R^4) = 84R^2 \quad \text{(Equation 4)} \]

### Step 5: Solve for \( A \)
From Equation 3, we can express \( A \):
\[ A = \frac{14R}{1 + R + R^2} \]
Substituting this expression for \( A \) into Equation 4:
\[ \left(\frac{14R}{1 + R + R^2}\right)^2(1 + R^2 + R^4) = 84R^2 \]
This simplifies to:
\[ \frac{196R^2(1 + R^2 + R^4)}{(1 + R + R^2)^2} = 84R^2 \]
Cancelling \( R^2 \) (assuming \( R \neq 0 \)):
\[ \frac{196(1 + R^2 + R^4)}{(1 + R + R^2)^2} = 84 \]

### Step 6: Cross-multiply and simplify
Cross-multiplying gives:
\[ 196(1 + R^2 + R^4) = 84(1 + R + R^2)^2 \]
Expanding both sides and simplifying will lead to a polynomial in \( R \).

### Step 7: Solve the polynomial
After simplification, you will get a quadratic equation in \( R \). Solve this quadratic equation using the quadratic formula or factoring.

### Step 8: Find values of \( A \) and \( R \)
Once you find \( R \), substitute back into the expression for \( A \) to find the first term. Then, compute the three terms in G.P. using \( A \) and \( R \).

### Step 9: Verify the conditions
Check if the sum of the three terms equals 14 and the sum of their squares equals 84.

### Final Result
The three numbers in G.P. are \( 2, 4, 8 \) or \( 8, 4, 2 \).

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