Find the sum of the following A.P.'s :
(i) `3,8,13, .... ` to 20 terms. (ii) `1,4,7,.........` to 50 terms.
(iii) `8,5,2, ....` to 25 terms. (iv) `(a+b), (2a+3b), (3b+5b), ........ ` to n terms.

Answer

To find the sum of the given arithmetic progressions (A.P.s), we will use the formula for the sum of the first \( n \) terms of an A.P.:

\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

Let's solve each part step-by-step.

### (i) For the A.P. \( 3, 8, 13, \ldots \) to 20 terms

1. **Identify the first term \( a \) and common difference \( d \)**:
- \( a = 3 \)
- \( d = 8 - 3 = 5 \)

2. **Substitute into the sum formula**:
\[ S_{20} = \frac{20}{2} \times (2 \times 3 + (20 - 1) \times 5) \]

3. **Calculate**:
\[ S_{20} = 10 \times (6 + 19 \times 5) \]
\[ = 10 \times (6 + 95) \]
\[ = 10 \times 101 = 1010 \]

### (ii) For the A.P. \( 1, 4, 7, \ldots \) to 50 terms

1. **Identify the first term \( a \) and common difference \( d \)**:
- \( a = 1 \)
- \( d = 4 - 1 = 3 \)

2. **Substitute into the sum formula**:
\[ S_{50} = \frac{50}{2} \times (2 \times 1 + (50 - 1) \times 3) \]

3. **Calculate**:
\[ S_{50} = 25 \times (2 + 49 \times 3) \]
\[ = 25 \times (2 + 147) \]
\[ = 25 \times 149 = 3725 \]

### (iii) For the A.P. \( 8, 5, 2, \ldots \) to 25 terms

1. **Identify the first term \( a \) and common difference \( d \)**:
- \( a = 8 \)
- \( d = 5 - 8 = -3 \)

2. **Substitute into the sum formula**:
\[ S_{25} = \frac{25}{2} \times (2 \times 8 + (25 - 1) \times (-3)) \]

3. **Calculate**:
\[ S_{25} = \frac{25}{2} \times (16 - 72) \]
\[ = \frac{25}{2} \times (-56) \]
\[ = 25 \times (-28) = -700 \]

### (iv) For the A.P. \( (a+b), (2a+3b), (3a+5b), \ldots \) to \( n \) terms

1. **Identify the first term \( a \) and common difference \( d \)**:
- First term \( a = a + b \)
- To find \( d \):
\[ d = (2a + 3b) - (a + b) = a + 2b \]

2. **Substitute into the sum formula**:
\[ S_n = \frac{n}{2} \times (2(a + b) + (n - 1)(a + 2b)) \]

3. **Simplify**:
\[ S_n = \frac{n}{2} \times (2a + 2b + (n - 1)(a + 2b)) \]
\[ = \frac{n}{2} \times (2a + 2b + (n - 1)a + 2(n - 1)b) \]
\[ = \frac{n}{2} \times (na + 2nb + a) \]
\[ = \frac{n}{2} \times ((n + 1)a + 2nb) \]

### Summary of Answers:
1. \( S_{20} = 1010 \)
2. \( S_{50} = 3725 \)
3. \( S_{25} = -700 \)
4. \( S_n = \frac{n}{2} \times ((n + 1)a + 2nb) \)