The sum of four terms in G.P. is 312 . The sum of first and fourth term is 252. Find the product of second and third term :

Options

A. 500
B. 1500
C. 60
D. none of these

Answer

To solve the problem step by step, let's denote the four terms of the geometric progression (G.P.) as follows:

1. First term: \( a \)
2. Second term: \( ar \)
3. Third term: \( ar^2 \)
4. Fourth term: \( ar^3 \)

### Step 1: Set up the equations based on the problem statement

From the problem, we know:

1. The sum of the four terms is 312:
\[ a + ar + ar^2 + ar^3 = 312 \]
Factoring out \( a \):
\[ a(1 + r + r^2 + r^3) = 312 \quad \text{(Equation 1)} \]

2. The sum of the first and fourth terms is 252:
\[ a + ar^3 = 252 \]
Factoring out \( a \):
\[ a(1 + r^3) = 252 \quad \text{(Equation 2)} \]

### Step 2: Divide Equation 1 by Equation 2

Now, we can divide Equation 1 by Equation 2 to eliminate \( a \):
\[ \frac{a(1 + r + r^2 + r^3)}{a(1 + r^3)} = \frac{312}{252} \]
This simplifies to:
\[ \frac{1 + r + r^2 + r^3}{1 + r^3} = \frac{312}{252} \]
Reducing \( \frac{312}{252} \):
\[ \frac{312 \div 12}{252 \div 12} = \frac{26}{21} \]
Thus, we have:
\[ \frac{1 + r + r^2 + r^3}{1 + r^3} = \frac{26}{21} \]

### Step 3: Cross-multiply to solve for \( r \)

Cross-multiplying gives:
\[ 21(1 + r + r^2 + r^3) = 26(1 + r^3) \]
Expanding both sides:
\[ 21 + 21r + 21r^2 + 21r^3 = 26 + 26r^3 \]
Rearranging terms:
\[ 21r + 21r^2 + 21r^3 - 26r^3 + 21 - 26 = 0 \]
This simplifies to:
\[ 21r + 21r^2 - 5r^3 - 5 = 0 \]
Dividing the entire equation by 5:
\[ - r^3 + \frac{21}{5}r^2 + \frac{21}{5}r - 1 = 0 \]

### Step 4: Solve the cubic equation

To solve for \( r \), we can use the Rational Root Theorem or numerical methods. However, we can also try simple rational values.

After testing, we find that \( r = 5 \) and \( r = \frac{1}{5} \) are the roots.

### Step 5: Find the value of \( a \)

Using \( r = 5 \) in Equation 2:
\[ a(1 + 5^3) = 252 \]
Calculating \( 5^3 = 125 \):
\[ a(1 + 125) = 252 \Rightarrow a(126) = 252 \Rightarrow a = \frac{252}{126} = 2 \]

### Step 6: Calculate the second and third terms

- Second term: \( ar = 2 \times 5 = 10 \)
- Third term: \( ar^2 = 2 \times 5^2 = 2 \times 25 = 50 \)

### Step 7: Find the product of the second and third terms

The product of the second and third terms:
\[ 10 \times 50 = 500 \]

### Final Answer

Thus, the product of the second and third terms is \( \boxed{500} \).

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