Define Geometrical progression.

Answer

### Step-by-Step Text Solution

1. **Definition of Geometric Progression (GP)**:
A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (denoted as \( R \)).

2. **Understanding the Sequence**:
In a GP, if the first term is \( a \), then the second term can be expressed as \( a \times R \), the third term as \( a \times R^2 \), the fourth term as \( a \times R^3 \), and so on. Thus, the \( n \)-th term of a GP can be represented as:
\[ a_n = a \times R^{(n-1)} \]

3. **Example of a GP**:
Consider the sequence: \( 1, 2, 4, 8, 16, 32, \ldots \)
- Here, the first term \( a = 1 \).
- The common ratio \( R \) can be calculated as:
- \( R = \frac{2}{1} = 2 \)
- \( R = \frac{4}{2} = 2 \)
- \( R = \frac{8}{4} = 2 \)
- \( R = \frac{16}{8} = 2 \)
- Thus, the common ratio \( R \) is consistently \( 2 \).

4. **Verifying the GP**:
To verify if a sequence is a GP, check if the ratio of consecutive terms is constant. For the example \( 1, 2, 4, 8, 16, 32 \):
- \( \frac{2}{1} = 2 \)
- \( \frac{4}{2} = 2 \)
- \( \frac{8}{4} = 2 \)
- \( \frac{16}{8} = 2 \)
- All ratios are equal to \( 2 \), confirming it is a GP.