If the third term of G.P.is `4`, then find the product of first five terms

Options

A. `4^(3)`
B. `4^(4)`
C. `4^(5)`
D. None of these

Explanation

To solve the problem step by step, we need to find the product of the first five terms of a geometric progression (G.P.) given that the third term is 4.

### Step-by-Step Solution:

1. **Understanding the Terms of G.P.**:
- In a geometric progression, the first term is denoted as \( a \) and the common ratio as \( r \).
- The terms of the G.P. can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)

2. **Identifying the Third Term**:
- We are given that the third term \( ar^2 = 4 \).

3. **Finding the Product of the First Five Terms**:
- The product of the first five terms can be calculated as follows:
\[ P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \]
- This simplifies to:
\[ P = a^5 \cdot r^{0+1+2+3+4} = a^5 \cdot r^{10} \]

4. **Expressing the Product in Terms of the Third Term**:
- We know from step 2 that \( ar^2 = 4 \).
- Therefore, we can express \( a \) in terms of \( ar^2 \):
\[ a = \frac{4}{r^2} \]
- Substituting this into the product:
\[ P = \left(\frac{4}{r^2}\right)^5 \cdot r^{10} \]
- This further simplifies to:
\[ P = \frac{4^5}{r^{10}} \cdot r^{10} = 4^5 \]

5. **Calculating \( 4^5 \)**:
- Now, we calculate \( 4^5 \):
\[ 4^5 = 1024 \]

6. **Final Answer**:
- Thus, the product of the first five terms of the G.P. is \( 1024 \).

### Final Answer:
The product of the first five terms is \( 1024 \).