Consider the following statements :
1. If each term of a GP is multiplied by same non-zero number, then the resulting sequence is also a GP.
2. If each term of a GP is divided by same non-zero number, then the resulting sequence is also a GP.
Which of the above statements is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To determine the correctness of the given statements regarding geometric progressions (GP), let's analyze each statement step by step.

### Step 1: Understanding a Geometric Progression (GP)
A geometric progression 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 (r). For example, if the first term is \( a \), the terms can be represented as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- Fourth term: \( ar^3 \)
- And so on...

### Step 2: Analyzing Statement 1
**Statement 1:** If each term of a GP is multiplied by the same non-zero number, then the resulting sequence is also a GP.

Let’s denote the GP as \( a, ar, ar^2, ar^3, \ldots \). If we multiply each term by a non-zero number \( k \), we get:
- New sequence: \( ka, kar, kar^2, kar^3, \ldots \)

The new sequence can be expressed as:
- First term: \( ka \)
- Second term: \( kar = ka \cdot r \)
- Third term: \( kar^2 = ka \cdot r^2 \)
- Fourth term: \( kar^3 = ka \cdot r^3 \)

Since the ratio between consecutive terms remains constant (which is \( r \)), the new sequence is also a GP.

**Conclusion for Statement 1:** This statement is **correct**.

### Step 3: Analyzing Statement 2
**Statement 2:** If each term of a GP is divided by the same non-zero number, then the resulting sequence is also a GP.

Using the same GP \( a, ar, ar^2, ar^3, \ldots \), if we divide each term by a non-zero number \( k \), we get:
- New sequence: \( \frac{a}{k}, \frac{ar}{k}, \frac{ar^2}{k}, \frac{ar^3}{k}, \ldots \)

The new sequence can be expressed as:
- First term: \( \frac{a}{k} \)
- Second term: \( \frac{ar}{k} = \frac{a}{k} \cdot r \)
- Third term: \( \frac{ar^2}{k} = \frac{a}{k} \cdot r^2 \)
- Fourth term: \( \frac{ar^3}{k} = \frac{a}{k} \cdot r^3 \)

Again, the ratio between consecutive terms remains constant (which is \( r \)), so the new sequence is also a GP.

**Conclusion for Statement 2:** This statement is **correct**.

### Final Conclusion
Both statements are correct. Therefore, the answer is that both statements 1 and 2 are correct.