The product of 5 terms of G.P. whose 3rd term is 2 is

Options

A. A. `5^(2)`
B. B. `2^(5)`
C. C. `3^(2)`
D. D. `3^(5)`

Answer

To find the product of 5 terms of a geometric progression (G.P.) whose third term is 2, we can follow these steps:

### Step 1: Understand the General Form of G.P.
The general term of a G.P. can be expressed as:
\[ T_n = A \cdot R^{n-1} \]
where:
- \( A \) is the first term,
- \( R \) is the common ratio,
- \( n \) is the term number.

### Step 2: Identify the Third Term
Given that the third term \( T_3 \) is 2, we can write:
\[ T_3 = A \cdot R^{3-1} = A \cdot R^2 = 2 \]

### Step 3: Write the First 5 Terms
The first five terms of the G.P. are:
- \( T_1 = A \)
- \( T_2 = A \cdot R \)
- \( T_3 = A \cdot R^2 \)
- \( T_4 = A \cdot R^3 \)
- \( T_5 = A \cdot R^4 \)

### Step 4: Calculate the Product of the First 5 Terms
The product \( P \) of the first five terms is:
\[ P = T_1 \cdot T_2 \cdot T_3 \cdot T_4 \cdot T_5 \]
Substituting the terms, we get:
\[ P = A \cdot (A \cdot R) \cdot (A \cdot R^2) \cdot (A \cdot R^3) \cdot (A \cdot R^4) \]
This simplifies to:
\[ P = A^5 \cdot R^{0 + 1 + 2 + 3 + 4} = A^5 \cdot R^{10} \]

### Step 5: Substitute \( A \cdot R^2 \) with \( T_3 \)
From Step 2, we know that \( A \cdot R^2 = 2 \). Therefore, we can express \( A \cdot R^2 \) as:
\[ A \cdot R^2 = 2 \]
Thus, we can rewrite \( A^5 \cdot R^{10} \) as:
\[ P = (A \cdot R^2)^{5} = 2^{5} \]

### Step 6: Final Result
The product of the first 5 terms of the G.P. is:
\[ P = 2^5 \]

### Conclusion
The correct answer is \( 2^5 \), which corresponds to option B.

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