The third term of a G.P. is 3. Find the product of its first five terms.

Options

A. 81
B. 243
C. 729
D. 343

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the properties of a Geometric Progression (G.P.)
In a G.P., the nth term can be expressed as:
\[ T_n = a \cdot r^{n-1} \]
where \( a \) is the first term and \( r \) is the common ratio.

### Step 2: Use the information given in the problem
We know that the third term of the G.P. is 3. Therefore, we can write:
\[ T_3 = a \cdot r^{3-1} = a \cdot r^2 = 3 \]
This gives us our first equation:
\[ a \cdot r^2 = 3 \quad \text{(1)} \]

### Step 3: Write the first five terms of the G.P.
The first five terms of the G.P. are:
1. First term: \( a \)
2. Second term: \( a \cdot r \)
3. Third term: \( a \cdot r^2 \)
4. Fourth term: \( a \cdot r^3 \)
5. Fifth term: \( a \cdot r^4 \)

### Step 4: Find the product of the first five terms
The product \( P \) of the first five terms can be calculated as follows:
\[ P = a \cdot (a \cdot r) \cdot (a \cdot r^2) \cdot (a \cdot r^3) \cdot (a \cdot r^4) \]
This simplifies to:
\[ P = a^5 \cdot r^{0+1+2+3+4} = a^5 \cdot r^{10} \]

### Step 5: Substitute \( a \cdot r^2 \) into the product
From equation (1), we know that \( a \cdot r^2 = 3 \). Therefore, we can express \( a^5 \cdot r^{10} \) in terms of \( a \cdot r^2 \):
\[ P = (a \cdot r^2)^{5} = 3^{5} \]

### Step 6: Calculate \( 3^5 \)
Now we compute \( 3^5 \):
\[ 3^5 = 243 \]

### Final Answer
Thus, the product of the first five terms of the G.P. is:
\[ \boxed{243} \]