When the product of a G.P. is known , then which are (i) three (ii) four (iii) five terms,

Answer

To solve the problem of finding the terms of a geometric progression (G.P.) when the product is known, we will derive the terms for three, four, and five terms step by step.

### Step 1: Three Terms of G.P.

1. **Let the three terms of the G.P. be** \( \frac{a}{r}, a, ar \).
- Here, \( a \) is the first term and \( r \) is the common ratio.

2. **Calculate the product of these terms**:
\[ \text{Product} = \left(\frac{a}{r}\right) \times a \times ar = \frac{a^3}{r} \]

3. **Set the product equal to K**:
\[ \frac{a^3}{r} = K \]

4. **From this, we can express \( a^3 \)**:
\[ a^3 = K \cdot r \]

### Step 2: Four Terms of G.P.

1. **Let the four terms of the G.P. be** \( \frac{a}{r^3}, \frac{a}{r}, a, ar^3 \).
- The first term is \( \frac{a}{r^3} \) and the common ratio is \( r^2 \).

2. **Calculate the product of these terms**:
\[ \text{Product} = \left(\frac{a}{r^3}\right) \times \left(\frac{a}{r}\right) \times a \times (ar^3) = \frac{a^4}{r^4} \]

3. **Set the product equal to K**:
\[ \frac{a^4}{r^4} = K \]

4. **From this, we can express \( a^4 \)**:
\[ a^4 = K \cdot r^4 \]

### Step 3: Five Terms of G.P.

1. **Let the five terms of the G.P. be** \( \frac{a}{r^2}, \frac{a}{r}, a, ar, ar^2 \).
- The first term is \( \frac{a}{r^2} \) and the common ratio is \( r \).

2. **Calculate the product of these terms**:
\[ \text{Product} = \left(\frac{a}{r^2}\right) \times \left(\frac{a}{r}\right) \times a \times (ar) \times (ar^2) = \frac{a^5}{r^5} \]

3. **Set the product equal to K**:
\[ \frac{a^5}{r^5} = K \]

4. **From this, we can express \( a^5 \)**:
\[ a^5 = K \cdot r^5 \]

### Summary of Results

- For **three terms**: \( a^3 = K \cdot r \)
- For **four terms**: \( a^4 = K \cdot r^4 \)
- For **five terms**: \( a^5 = K \cdot r^5 \)