Out of l,a,n,d and `S_(n)`, determine the one that are missing from each of the following:
(i) `l=8,n=8,S_(8)=-20`
(ii) `a=-3030,l=-1530,n=51`

Answer

To solve the problem, we will analyze each part step by step.

### Part (i): Given \( l = 8, n = 8, S_8 = -20 \)

1. **Identify the formula for the sum of the first n terms of an arithmetic progression (AP)**:
\[ S_n = \frac{n}{2} \times (A + L) \]
where \( S_n \) is the sum of the first n terms, \( A \) is the first term, \( L \) is the last term, and \( n \) is the number of terms.

2. **Substitute the known values into the formula**:
\[ S_8 = \frac{8}{2} \times (A + 8) = -20 \]
This simplifies to:
\[ 4(A + 8) = -20 \]

3. **Solve for \( A \)**:
\[ A + 8 = \frac{-20}{4} = -5 \]
\[ A = -5 - 8 = -13 \]

4. **Now find the common difference \( D \)** using the nth term formula:
\[ A_n = A + (n - 1)D \]
Here, \( A_8 = 8 \):
\[ 8 = -13 + (8 - 1)D \]
\[ 8 = -13 + 7D \]
\[ 7D = 8 + 13 = 21 \]
\[ D = \frac{21}{7} = 3 \]

**Conclusion for Part (i)**:
- The missing term is \( A = -13 \) and \( D = 3 \).

### Part (ii): Given \( A = -3030, L = -1530, n = 51 \)

1. **Use the same formula for the sum of the first n terms**:
\[ S_n = \frac{n}{2} \times (A + L) \]
Substitute the known values:
\[ S_{51} = \frac{51}{2} \times (-3030 - 1530) \]
\[ S_{51} = \frac{51}{2} \times (-4560) \]

2. **Calculate \( S_{51} \)**:
\[ S_{51} = \frac{51 \times -4560}{2} \]
\[ S_{51} = 51 \times -2280 = -116280 \]

3. **Now find the common difference \( D \)**:
Using the nth term formula:
\[ A_n = A + (n - 1)D \]
Here, \( A_{51} = L = -1530 \):
\[ -1530 = -3030 + (51 - 1)D \]
\[ -1530 = -3030 + 50D \]
\[ 50D = -1530 + 3030 = 1500 \]
\[ D = \frac{1500}{50} = 30 \]

**Conclusion for Part (ii)**:
- The missing term is \( S_{51} = -116280 \) and \( D = 30 \).

### Final Answers:
- Part (i): \( A = -13 \), \( D = 3 \)
- Part (ii): \( S_{51} = -116280 \), \( D = 30 \)