The third term of a GP is 3. What is the product of the first five terms?

Options

A. 216
B. 226
C. 243
D. Cannot be determined due to insuffcient data

Explanation

To find the product of the first five terms of a geometric progression (GP) given that the third term is 3, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the terms of the GP**:
In a GP, the terms can be expressed as follows:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- Fifth term: \( ar^4 \)

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

3. **Write the product of the first five terms**:
The product of the first five terms \( P \) is given by:
\[ P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \]

4. **Simplify the product**:
\[ P = a^5 \cdot r^{0 + 1 + 2 + 3 + 4} = a^5 \cdot r^{10} \]

5. **Express \( a \) in terms of \( r \)**:
From the third term, we have:
\[ ar^2 = 3 \implies a = \frac{3}{r^2} \]

6. **Substitute \( a \) into the product**:
Substitute \( a \) in the product expression:
\[ P = \left(\frac{3}{r^2}\right)^5 \cdot r^{10} \]
\[ P = \frac{3^5}{r^{10}} \cdot r^{10} \]
\[ P = 3^5 \]

7. **Calculate \( 3^5 \)**:
\[ 3^5 = 243 \]

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

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