(a) Which term of the progression `10,9(1)/(3),8 (2)/(3),`...is the first negative term ?
(b) Which term of the progression `4,3(5)/(7),3(3)/(7),`...is the first negative term ?

Explanation

To solve the given problem, we will break it down into two parts as stated in the question.

### Part (a): Finding the first negative term of the progression `10, 9(1)/(3), 8(2)/(3), ...`

1. **Convert mixed fractions to improper fractions**:
- The first term is \(10\).
- The second term \(9 \frac{1}{3} = \frac{28}{3}\).
- The third term \(8 \frac{2}{3} = \frac{26}{3}\).
- Thus, the sequence can be written as \(10, \frac{28}{3}, \frac{26}{3}, ...\).

2. **Identify the first term (a) and common difference (d)**:
- The first term \(a = 10\).
- To find the common difference \(d\):
\[ d = \frac{28}{3} - 10 = \frac{28}{3} - \frac{30}{3} = \frac{-2}{3} \]

3. **Write the general term (a_n)**:
- The general term of an arithmetic progression is given by:
\[ a_n = a + (n-1)d \]
- Substituting the values of \(a\) and \(d\):
\[ a_n = 10 + (n-1)\left(-\frac{2}{3}\right) = 10 - \frac{2(n-1)}{3} \]
- Simplifying:
\[ a_n = 10 - \frac{2n - 2}{3} = \frac{30}{3} - \frac{2n - 2}{3} = \frac{32 - 2n}{3} \]

4. **Find when the general term is negative**:
- Set \(a_n < 0\):
\[ \frac{32 - 2n}{3} < 0 \]
- Multiply through by 3 (since 3 is positive):
\[ 32 - 2n < 0 \]
- Rearranging gives:
\[ 2n > 32 \implies n > 16 \]

5. **Determine the first negative term**:
- The first integer \(n\) greater than 16 is \(n = 17\).
- Thus, the first negative term is the **17th term**.

### Part (b): Finding the first negative term of the progression `4, 3(5)/(7), 3(3)/(7), ...`

1. **Convert mixed fractions to improper fractions**:
- The first term is \(4\).
- The second term \(3 \frac{5}{7} = \frac{26}{7}\).
- The third term \(3 \frac{3}{7} = \frac{24}{7}\).
- Thus, the sequence can be written as \(4, \frac{26}{7}, \frac{24}{7}, ...\).

2. **Identify the first term (a) and common difference (d)**:
- The first term \(a = 4\).
- To find the common difference \(d\):
\[ d = \frac{26}{7} - 4 = \frac{26}{7} - \frac{28}{7} = \frac{-2}{7} \]

3. **Write the general term (a_n)**:
- The general term of an arithmetic progression is given by:
\[ a_n = a + (n-1)d \]
- Substituting the values of \(a\) and \(d\):
\[ a_n = 4 + (n-1)\left(-\frac{2}{7}\right) = 4 - \frac{2(n-1)}{7} \]
- Simplifying:
\[ a_n = 4 - \frac{2n - 2}{7} = \frac{28}{7} - \frac{2n - 2}{7} = \frac{30 - 2n}{7} \]

4. **Find when the general term is negative**:
- Set \(a_n < 0\):
\[ \frac{30 - 2n}{7} < 0 \]
- Multiply through by 7 (since 7 is positive):
\[ 30 - 2n < 0 \]
- Rearranging gives:
\[ 2n > 30 \implies n > 15 \]

5. **Determine the first negative term**:
- The first integer \(n\) greater than 15 is \(n = 16\).
- Thus, the first negative term is the **16th term**.

### Summary of Answers:
- (a) The first negative term of the progression is the **17th term**.
- (b) The first negative term of the progression is the **16th term**.