The third term of G.P. is 4. The product of its first 5 terms is -

Options

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

Answer

To solve the problem, 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. **Identify the Terms of the G.P.:**
In a G.P., the first term is denoted as \( a \) and the common ratio as \( r \). The first five 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. **Use the Given Information:**
We know that the third term \( ar^2 = 4 \).

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

4. **Express \( a \) in Terms of \( r \):**
From the equation \( ar^2 = 4 \), we can express \( a \) as:
\[ a = \frac{4}{r^2} \]

5. **Substitute \( a \) into the Product Formula:**
Now substitute \( a \) into the product \( P \):
\[ P = \left(\frac{4}{r^2}\right)^5 \cdot r^{10} \]
Simplifying this gives:
\[ P = \frac{4^5}{r^{10}} \cdot r^{10} = 4^5 \]

6. **Calculate \( 4^5 \):**
Now we calculate \( 4^5 \):
\[ 4^5 = 1024 \]

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

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