We are given two G.P's one with the first term a and common ratio r and the other with first term b and common ratio s. Show that the sequence formed by the product of corresponding terms is a G.P. Find its first term and the common ratio. Show also that the sequence formed by the quotient of corresponding terms is in G.P. Find its first term and common ratio.

Answer

To solve the problem, we need to show that the sequences formed by the product and the quotient of the corresponding terms of two geometric progressions (G.P.s) are also G.P.s. We will find their first terms and common ratios.

### Step 1: Define the two G.P.s

Let the first G.P. have:
- First term = \( a \)
- Common ratio = \( r \)

The terms of this G.P. are:
- \( a, ar, ar^2, ar^3, \ldots \)

Let the second G.P. have:
- First term = \( b \)
- Common ratio = \( s \)

The terms of this G.P. are:
- \( b, bs, bs^2, bs^3, \ldots \)

### Step 2: Form the sequence of products

The sequence formed by the product of the corresponding terms of the two G.P.s is:
- \( ab, ar \cdot bs, ar^2 \cdot bs^2, ar^3 \cdot bs^3, \ldots \)

Calculating the terms:
1. First term: \( ab \)
2. Second term: \( ar \cdot bs = abrs \)
3. Third term: \( ar^2 \cdot bs^2 = abrs^2 \)
4. Fourth term: \( ar^3 \cdot bs^3 = abrs^3 \)

Thus, the sequence of products is:
- \( ab, abrs, abrs^2, abrs^3, \ldots \)

### Step 3: Show that the product sequence is a G.P.

To show that this sequence is a G.P., we need to find the common ratio.

Calculating the ratio of consecutive terms:
1. The ratio of the second term to the first term:
\[ R_1 = \frac{abrs}{ab} = rs \]

2. The ratio of the third term to the second term:
\[ R_2 = \frac{abrs^2}{abrs} = s \]

3. The ratio of the fourth term to the third term:
\[ R_3 = \frac{abrs^3}{abrs^2} = s \]

Since \( R_1 = R_2 = R_3 = rs \), the common ratio of the product sequence is \( rs \).

### Step 4: Find the first term and common ratio of the product sequence

- First term = \( ab \)
- Common ratio = \( rs \)

### Step 5: Form the sequence of quotients

The sequence formed by the quotient of the corresponding terms of the two G.P.s is:
- \( \frac{a}{b}, \frac{ar}{bs}, \frac{ar^2}{bs^2}, \frac{ar^3}{bs^3}, \ldots \)

Calculating the terms:
1. First term: \( \frac{a}{b} \)
2. Second term: \( \frac{ar}{bs} = \frac{a}{b} \cdot \frac{r}{s} \)
3. Third term: \( \frac{ar^2}{bs^2} = \frac{a}{b} \cdot \frac{r^2}{s^2} \)
4. Fourth term: \( \frac{ar^3}{bs^3} = \frac{a}{b} \cdot \frac{r^3}{s^3} \)

Thus, the sequence of quotients is:
- \( \frac{a}{b}, \frac{a}{b} \cdot \frac{r}{s}, \frac{a}{b} \cdot \frac{r^2}{s^2}, \frac{a}{b} \cdot \frac{r^3}{s^3}, \ldots \)

### Step 6: Show that the quotient sequence is a G.P.

To show that this sequence is a G.P., we need to find the common ratio.

Calculating the ratio of consecutive terms:
1. The ratio of the second term to the first term:
\[ R_1 = \frac{\frac{ar}{bs}}{\frac{a}{b}} = \frac{r}{s} \]

2. The ratio of the third term to the second term:
\[ R_2 = \frac{\frac{ar^2}{bs^2}}{\frac{ar}{bs}} = \frac{r^2}{s^2} \cdot \frac{bs}{ar} = \frac{r}{s} \]

3. The ratio of the fourth term to the third term:
\[ R_3 = \frac{\frac{ar^3}{bs^3}}{\frac{ar^2}{bs^2}} = \frac{r^3}{s^3} \cdot \frac{bs^2}{ar^2} = \frac{r}{s} \]

Since \( R_1 = R_2 = R_3 = \frac{r}{s} \), the common ratio of the quotient sequence is \( \frac{r}{s} \).

### Step 7: Find the first term and common ratio of the quotient sequence

- First term = \( \frac{a}{b} \)
- Common ratio = \( \frac{r}{s} \)

### Summary of Results:
- For the product sequence:
- First term = \( ab \)
- Common ratio = \( rs \)

- For the quotient sequence:
- First term = \( \frac{a}{b} \)
- Common ratio = \( \frac{r}{s} \)