Find r and write the next four terms of each of the following progresssion:
(i) `5,0.5,0.05………….` (ii) `-2/3,-6,-54`…………….

Answer

To solve the problem, we need to find the common ratio \( r \) for each progression and then write the next four terms.

### (i) Progression: \( 5, 0.5, 0.05, \ldots \)

**Step 1: Find the common ratio \( r \)**
The common ratio \( r \) can be calculated using the formula:
\[ r = \frac{a_n}{a_{n-1}} \]
Taking the first two terms:
\[ r = \frac{0.5}{5} = \frac{1}{10} \]

**Step 2: Write the next four terms**
The first term \( a = 5 \) and the common ratio \( r = \frac{1}{10} \). The \( n \)-th term of a geometric progression is given by:
\[ a_n = a \cdot r^{n-1} \]
Calculating the next four terms:
- 3rd term: \( a_3 = 5 \cdot \left(\frac{1}{10}\right)^{2} = 5 \cdot \frac{1}{100} = 0.05 \)
- 4th term: \( a_4 = 5 \cdot \left(\frac{1}{10}\right)^{3} = 5 \cdot \frac{1}{1000} = 0.005 \)
- 5th term: \( a_5 = 5 \cdot \left(\frac{1}{10}\right)^{4} = 5 \cdot \frac{1}{10000} = 0.0005 \)
- 6th term: \( a_6 = 5 \cdot \left(\frac{1}{10}\right)^{5} = 5 \cdot \frac{1}{100000} = 0.00005 \)

Thus, the next four terms are:
\[ 0.005, 0.0005, 0.00005, 0.000005 \]

### (ii) Progression: \( -\frac{2}{3}, -6, -54, \ldots \)

**Step 1: Find the common ratio \( r \)**
Using the same formula:
\[ r = \frac{a_n}{a_{n-1}} \]
Taking the first two terms:
\[ r = \frac{-6}{-\frac{2}{3}} = -6 \cdot \left(-\frac{3}{2}\right) = 9 \]

**Step 2: Write the next four terms**
The first term \( a = -\frac{2}{3} \) and the common ratio \( r = 9 \). Calculating the next four terms:
- 4th term: \( a_4 = -54 \cdot 9 = -486 \)
- 5th term: \( a_5 = -486 \cdot 9 = -4374 \)
- 6th term: \( a_6 = -4374 \cdot 9 = -39366 \)
- 7th term: \( a_7 = -39366 \cdot 9 = -354294 \)

Thus, the next four terms are:
\[ -486, -4374, -39366, -354294 \]

### Final Answers:
- For the first progression: \( 0.005, 0.0005, 0.00005, 0.000005 \)
- For the second progression: \( -486, -4374, -39366, -354294 \)