The third term of a GP is 3. what is the product of the first 5 terms ?

Options

A. A)216
B. B)226
C. C)243
D. D)Can not determined

Answer

To solve the problem, we need to find the product of the first five terms of a geometric progression (GP) where the third term is given as 3.

### Step-by-step Solution:

1. **Identify the terms of the GP**:
- Let the first term be \( a \).
- The second term will be \( ar \) (where \( r \) is the common ratio).
- The third term is \( ar^2 \).
- The fourth term is \( ar^3 \).
- The fifth term is \( ar^4 \).

2. **Use the given information**:
- We know that the third term \( ar^2 = 3 \).

3. **Express the terms**:
- The first term is \( a \).
- The second term is \( ar \).
- The third term is \( ar^2 = 3 \).
- The fourth term is \( ar^3 \).
- The fifth term is \( ar^4 \).

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

5. **Express \( r^{10} \) in terms of \( a \)**:
- From the equation \( ar^2 = 3 \), we can express \( r^2 \) as:
\[ r^2 = \frac{3}{a} \]
- Therefore, \( r^{10} = (r^2)^5 = \left(\frac{3}{a}\right)^5 = \frac{243}{a^5} \).

6. **Substitute \( r^{10} \) back into the product**:
- Now substituting \( r^{10} \) into the product \( P \):
\[ P = a^5 \cdot \frac{243}{a^5} = 243 \]

### Final Answer:
The product of the first five terms of the GP is **243**.