In a G.P. the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of infinite terms of the G.P is

Options

A. `-8`
B. `-8/3`
C. `8/3`
D. 8

Answer

To solve the problem, we need to find the sum of the infinite terms of a geometric progression (G.P.) given the conditions about the first four terms and their relationships. Let's break it down step-by-step.

### Step 1: Define the terms of the G.P.
Let the first term of the G.P. be \( a \) and the common ratio be \( r \). The first four terms of the G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)

### Step 2: Use the product of the first four terms
We know that the product of the first four terms is 4:
\[ a \cdot ar \cdot ar^2 \cdot ar^3 = 4 \]
This simplifies to:
\[ a^4 r^6 = 4 \]

### Step 3: Use the relationship between the second and fourth terms
We are also given that the second term is the reciprocal of the fourth term:
\[ ar = \frac{1}{ar^3} \]
Multiplying both sides by \( ar^3 \) gives:
\[ (ar)(ar^3) = 1 \implies a^2 r^4 = 1 \]

### Step 4: Solve the equations
Now we have two equations:
1. \( a^4 r^6 = 4 \)
2. \( a^2 r^4 = 1 \)

From the second equation, we can express \( a^2 \) in terms of \( r \):
\[ a^2 = \frac{1}{r^4} \]
Taking the square root gives:
\[ a = \frac{1}{r^2} \quad \text{(considering the positive root for simplicity)} \]

### Step 5: Substitute \( a \) into the first equation
Substituting \( a = \frac{1}{r^2} \) into the first equation:
\[ \left(\frac{1}{r^2}\right)^4 r^6 = 4 \]
This simplifies to:
\[ \frac{1}{r^8} r^6 = 4 \implies \frac{1}{r^2} = 4 \implies r^2 = \frac{1}{4} \implies r = \frac{1}{2} \quad \text{(taking the positive root)} \]

### Step 6: Find \( a \)
Now substituting \( r = \frac{1}{2} \) back into \( a = \frac{1}{r^2} \):
\[ a = \frac{1}{\left(\frac{1}{2}\right)^2} = \frac{1}{\frac{1}{4}} = 4 \]

### Step 7: Calculate the sum of infinite terms
The sum of the infinite terms of a G.P. is given by the formula:
\[ S = \frac{a}{1 - r} \]
Substituting \( a = 4 \) and \( r = \frac{1}{2} \):
\[ S = \frac{4}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 4 \times 2 = 8 \]

### Final Answer
The sum of the infinite terms of the G.P. is \( \boxed{8} \).