If fifth term of a G.P. is 2, then the product of its first 9 terms is

Options

A. 256
B. 512
C. 1024
D. none of these

Answer

To find the product of the first 9 terms of a geometric progression (G.P.) where the fifth term is given as 2, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the fifth term of the G.P.**:
The fifth term of a G.P. can be expressed as:
\[ T_5 = ar^4 = 2 \]
where \( a \) is the first term and \( r \) is the common ratio.

2. **Write the product of the first 9 terms**:
The first 9 terms of the G.P. are:
\[ a, ar, ar^2, ar^3, ar^4, ar^5, ar^6, ar^7, ar^8 \]
The product \( P \) of these terms can be expressed as:
\[ P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \cdot ar^5 \cdot ar^6 \cdot ar^7 \cdot ar^8 \]

3. **Factor out \( a \) and \( r \)**:
This can be simplified to:
\[ P = a^9 \cdot (r^0 + r^1 + r^2 + r^3 + r^4 + r^5 + r^6 + r^7 + r^8) \]
The exponent of \( r \) can be calculated as:
\[ 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \frac{8(8 + 1)}{2} = 36 \]
Therefore, we have:
\[ P = a^9 \cdot r^{36} \]

4. **Express \( a \) in terms of \( T_5 \)**:
From the equation \( ar^4 = 2 \), we can express \( a \) as:
\[ a = \frac{2}{r^4} \]

5. **Substitute \( a \) into the product formula**:
Substitute \( a \) back into the product:
\[ P = \left(\frac{2}{r^4}\right)^9 \cdot r^{36} \]
Simplifying this gives:
\[ P = \frac{2^9}{r^{36}} \cdot r^{36} = 2^9 \]

6. **Calculate \( 2^9 \)**:
Finally, calculate \( 2^9 \):
\[ 2^9 = 512 \]

### Final Answer:
Thus, the product of the first 9 terms of the G.P. is:
\[ \boxed{512} \]