Find the sum of :
(i) all odd natural numbers less than 50.
(ii) first 12 natural numbers each of which is a multiple of 7.

Answer

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

### Part (i): Find the sum of all odd natural numbers less than 50.

1. **Identify the sequence of odd natural numbers less than 50:**
The odd natural numbers less than 50 are: 1, 3, 5, 7, ..., 49.

2. **Determine the first term (a) and the last term (a_n):**
- First term (a) = 1
- Last term (a_n) = 49

3. **Identify the common difference (d):**
The common difference between consecutive odd numbers is 2 (e.g., 3 - 1 = 2, 5 - 3 = 2, etc.).

4. **Find the number of terms (n) in the sequence:**
We can use the formula for the nth term of an arithmetic progression:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 49 = 1 + (n - 1) \cdot 2 \]
Rearranging gives:
\[ 49 - 1 = (n - 1) \cdot 2 \implies 48 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 24 = n - 1 \implies n = 25 \]

5. **Calculate the sum (S_n) of the first n terms:**
The formula for the sum of the first n terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \cdot (a + a_n) \]
Substituting the values we found:
\[ S_{25} = \frac{25}{2} \cdot (1 + 49) = \frac{25}{2} \cdot 50 \]
Simplifying:
\[ S_{25} = 25 \cdot 25 = 625 \]

### Part (ii): Find the sum of the first 12 natural numbers that are multiples of 7.

1. **Identify the sequence of multiples of 7:**
The first 12 multiples of 7 are: 7, 14, 21, ..., 84.

2. **Determine the first term (a) and the last term (a_n):**
- First term (a) = 7
- Last term (a_n) = 84 (which is 7 × 12)

3. **Identify the common difference (d):**
The common difference is 7 (e.g., 14 - 7 = 7, 21 - 14 = 7, etc.).

4. **Identify the number of terms (n):**
Here, n is given as 12.

5. **Calculate the sum (S_n) of the first n terms:**
Using the sum formula:
\[ S_n = \frac{n}{2} \cdot (a + a_n) \]
Substituting the values:
\[ S_{12} = \frac{12}{2} \cdot (7 + 84) = 6 \cdot 91 \]
Simplifying:
\[ S_{12} = 546 \]

### Final Answers:
- (i) The sum of all odd natural numbers less than 50 is **625**.
- (ii) The sum of the first 12 natural numbers that are multiples of 7 is **546**.