What is the product of first `2n+1` terms of a geometric progression ?

Options

A. The `(n+1)th` power of the nth term of the GP
B. The `(2n+1)th` power of the nth term of the GP
C. The `(2n+1)th` power of the `(n+1)th` term of the GP
D. The nth power of the `(n+1)th` terms of the GP

Explanation

To find the product of the first \(2n + 1\) terms of a geometric progression (GP), we can follow these steps:

### Step-by-Step Solution:

1. **Define the GP**:
Let the first term of the geometric progression be \(A\) and the common ratio be \(R\). The terms of the GP can be expressed as:
\[ A, AR, AR^2, AR^3, \ldots, AR^{2n} \]
Thus, the first \(2n + 1\) terms of the GP are \(A, AR, AR^2, \ldots, AR^{2n}\).

2. **Write the Product**:
The product \(P\) of the first \(2n + 1\) terms can be written as:
\[ P = A \cdot AR \cdot AR^2 \cdot AR^3 \cdots AR^{2n} \]

3. **Factor Out Common Terms**:
We can factor out \(A\) from each term:
\[ P = A^{2n + 1} \cdot (R^0 \cdot R^1 \cdot R^2 \cdots R^{2n}) \]

4. **Simplify the Product of R Terms**:
The product of the \(R\) terms can be simplified. The exponents of \(R\) form an arithmetic series:
\[ 0 + 1 + 2 + \ldots + 2n = \frac{(2n)(2n + 1)}{2} = n(2n + 1) \]
Therefore, we have:
\[ P = A^{2n + 1} \cdot R^{n(2n + 1)} \]

5. **Final Expression**:
Thus, the product of the first \(2n + 1\) terms of the geometric progression is:
\[ P = A^{2n + 1} \cdot R^{n(2n + 1)} \]

### Conclusion:
The product of the first \(2n + 1\) terms of a geometric progression is given by:
\[ P = (AR^n)^{2n + 1} \]
This means that the product is equal to the \((n + 1)\)th term of the GP raised to the power of \(2n + 1\).