If the 5th term of a GP is 2, find the product of its first nine terms.

Explanation

To solve the problem step by step, we will follow the concepts of Geometric Progression (GP).

### Step 1: Understand the 5th term of a GP
The formula for the nth term of a GP is given by:
\[ T_n = A \cdot R^{n-1} \]
where \( A \) is the first term, \( R \) is the common ratio, and \( n \) is the term number.

Given that the 5th term \( T_5 = 2 \), we can write:
\[ T_5 = A \cdot R^{5-1} = A \cdot R^4 = 2 \]

### Step 2: Find the product of the first 9 terms
The product of the first \( n \) terms of a GP can be calculated using the formula:
\[ P_n = A^n \cdot R^{\frac{n(n-1)}{2}} \]
For the first 9 terms, we have \( n = 9 \):
\[ P_9 = A^9 \cdot R^{\frac{9(9-1)}{2}} = A^9 \cdot R^{36} \]

### Step 3: Express \( R^{36} \) in terms of \( A \) and \( R^4 \)
From the previous step, we know that \( R^4 = \frac{2}{A} \). To find \( R^{36} \), we can express it as:
\[ R^{36} = (R^4)^9 = \left(\frac{2}{A}\right)^9 \]

### Step 4: Substitute \( R^{36} \) back into the product formula
Now substituting \( R^{36} \) into the product formula:
\[ P_9 = A^9 \cdot \left(\frac{2}{A}\right)^9 \]
This simplifies to:
\[ P_9 = A^9 \cdot \frac{2^9}{A^9} \]
\[ P_9 = 2^9 \]

### Step 5: Calculate \( 2^9 \)
Now we can calculate \( 2^9 \):
\[ 2^9 = 512 \]

### Final Answer
Thus, the product of the first nine terms of the GP is:
\[ \boxed{512} \]

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