Key Concepts and Formulas

• Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The $k$-th term is given by $T_k = ar^{k-1}$, where $a$ is the first term.
• Product of $n$ terms of a GP ($P_n$): The product of the first $n$ terms is $P_n = a^n r^{\frac{n(n-1)}{2}}$.
• Special Property for Odd Number of Terms: If $n$ is odd, the product of the first $n$ terms of a GP is equal to the middle term raised to the power of $n$. The middle term is $T_{(n+1)/2}$. Thus, $P_n = (T_{(n+1)/2})^n$.

Step-by-Step Solution

Step 1: Identify the total number of terms and the position of the middle term. We are asked to find the product of 9 terms of a GP. So, $n=9$. Since $n=9$ is an odd number, there is a unique middle term. The position of the middle term in a sequence of $n$ terms is given by $\frac{n+1}{2}$. For $n=9$, the middle term is at position $\frac{9+1}{2} = \frac{10}{2} = 5$. Reasoning: Identifying the middle term is crucial for applying the special property of GPs, which simplifies the problem significantly.

Step 2: Relate the given information to the middle term. We are given that the fifth term of the GP is 2. This means $T_5 = 2$. From Step 1, we know that the 5th term is indeed the middle term for a sequence of 9 terms. Therefore, the middle term is 2. Reasoning: This step directly connects the given value to the concept of the middle term, which is central to the efficient solution method.

Step 3: Apply the special property for the product of an odd number of terms. The special property states that for an odd number of terms ($n$), the product of the terms ($P_n$) is equal to the middle term raised to the power of $n$. In this case, $n=9$ and the middle term is $T_5$. So, the product of the 9 terms, $P_9$, is given by: $P_9 = (T_5)^9$ Reasoning: This is the core principle that allows us to solve the problem without needing to find the first term ($a$) or the common ratio ($r$) explicitly.

Step 4: Substitute the given value and calculate the product. We are given $T_5 = 2$. Substituting this value into the formula from Step 3: $P_9 = (2)^9$ Now, we calculate the value of $2^9$: $2^1 = 2$ $2^2 = 4$ $2^3 = 8$ $2^4 = 16$ $2^5 = 32$ $2^6 = 64$ $2^7 = 128$ $2^8 = 256$ $2^9 = 512$ So, $P_9 = 512$. Reasoning: This is the final calculation step to arrive at the numerical answer.

Common Mistakes & Tips

• Forgetting the Middle Term Property: Many students might try to find $a$ and $r$ first. While possible, it's more time-consuming and prone to errors. Always look for the middle term property when $n$ is odd.
• Miscalculating Exponents: Ensure accuracy when calculating powers, especially for larger numbers like $2^9$.
• Confusing Product and Sum: Be clear whether the question asks for the sum or product of terms. The formulas are distinct.

Summary

The problem asks for the product of the first 9 terms of a Geometric Progression, given that the 5th term is 2. Since the number of terms (9) is odd, we can utilize the property that the product of these terms is equal to the middle term raised to the power of the number of terms. The 5th term is indeed the middle term for a sequence of 9 terms. Therefore, the product of the 9 terms is $(T_5)^9 = 2^9 = 512$.

The final answer is $\boxed{512}$.