The sum of first 20 odd natural numbers is equal to :

Options

A. 210
B. 300
C. 400
D. 420

Answer

To find the sum of the first 20 odd natural numbers, we can follow these steps:

### Step 1: Identify the sequence of odd natural numbers
The first odd natural numbers are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39

### Step 2: Recognize the pattern
This sequence is an arithmetic progression (AP) where:
- The first term \( A = 1 \)
- The common difference \( D = 3 - 1 = 2 \)

### Step 3: Find the 20th odd natural number
To find the 20th term of the sequence, we can use the formula for the nth term of an AP:
\[ A_n = A + (n - 1) \cdot D \]
Substituting the values:
- \( A = 1 \)
- \( n = 20 \)
- \( D = 2 \)

Calculating:
\[ A_{20} = 1 + (20 - 1) \cdot 2 = 1 + 19 \cdot 2 = 1 + 38 = 39 \]

### Step 4: Calculate the sum of the first 20 odd natural numbers
The formula for the sum \( S_n \) of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \cdot (A + A_n) \]
Where:
- \( n = 20 \)
- \( A = 1 \)
- \( A_n = 39 \)

Substituting the values:
\[ S_{20} = \frac{20}{2} \cdot (1 + 39) = 10 \cdot 40 = 400 \]

### Conclusion
The sum of the first 20 odd natural numbers is \( 400 \).

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