(i) Find the number of terms in the A.P.
8, 12, 16, ........124
(ii) Find the number of terms in the A.P.
75, 70, 65, .....15

Answer

To solve the problem, we need to find the number of terms in two arithmetic progressions (A.P.).

### Part (i): Find the number of terms in the A.P. 8, 12, 16, ..., 124

1. **Identify the first term (A)**:
- The first term \( A = 8 \).

2. **Find the common difference (D)**:
- The common difference \( D = 12 - 8 = 4 \).

3. **Identify the last term (An)**:
- The last term \( An = 124 \).

4. **Use the formula for the nth term of an A.P.**:
- The formula is given by:
\[ An = A + (n - 1)D \]
- Substitute the known values into the formula:
\[ 124 = 8 + (n - 1) \cdot 4 \]

5. **Simplify the equation**:
- Rearranging gives:
\[ 124 = 8 + 4n - 4 \]
- This simplifies to:
\[ 124 = 4 + 4n \]

6. **Isolate the term with n**:
- Subtract 4 from both sides:
\[ 120 = 4n \]

7. **Solve for n**:
- Divide both sides by 4:
\[ n = \frac{120}{4} = 30 \]

8. **Conclusion**:
- The number of terms in the A.P. is **30**.

### Part (ii): Find the number of terms in the A.P. 75, 70, 65, ..., 15

1. **Identify the first term (A)**:
- The first term \( A = 75 \).

2. **Find the common difference (D)**:
- The common difference \( D = 70 - 75 = -5 \).

3. **Identify the last term (An)**:
- The last term \( An = 15 \).

4. **Use the formula for the nth term of an A.P.**:
- The formula is:
\[ An = A + (n - 1)D \]
- Substitute the known values into the formula:
\[ 15 = 75 + (n - 1)(-5) \]

5. **Simplify the equation**:
- Rearranging gives:
\[ 15 = 75 - 5n + 5 \]
- This simplifies to:
\[ 15 = 80 - 5n \]

6. **Isolate the term with n**:
- Subtract 80 from both sides:
\[ -65 = -5n \]

7. **Solve for n**:
- Divide both sides by -5:
\[ n = \frac{65}{5} = 13 \]

8. **Conclusion**:
- The number of terms in the A.P. is **13**.

### Summary of Answers:
- (i) The number of terms in the A.P. 8, 12, 16, ..., 124 is **30**.
- (ii) The number of terms in the A.P. 75, 70, 65, ..., 15 is **13**.