Find the sum of first 21 odd numbers.

Options

A. 213
B. 231
C. 441
D. 144

Answer

To find the sum of the first 21 odd numbers, we can follow these steps:

### Step 1: Identify the first 21 odd numbers
The sequence of odd numbers starts from 1 and continues as follows:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41.

### Step 2: Recognize that this is an arithmetic progression (AP)
The first term (a) is 1, and the common difference (d) is 2.

### Step 3: Find the 21st odd number
To find the nth term of an AP, we use the formula:
\[ a_n = a + (n - 1) \cdot d \]
Here, \( n = 21 \), \( a = 1 \), and \( d = 2 \).
Substituting these values:
\[ a_{21} = 1 + (21 - 1) \cdot 2 \]
\[ a_{21} = 1 + 20 \cdot 2 \]
\[ a_{21} = 1 + 40 \]
\[ a_{21} = 41 \]

### Step 4: Calculate the sum of the first 21 odd numbers
The sum \( S_n \) of the first n terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
where \( l \) is the last term. Here, \( n = 21 \), \( a = 1 \), and \( l = 41 \).
Substituting these values:
\[ S_{21} = \frac{21}{2} \cdot (1 + 41) \]
\[ S_{21} = \frac{21}{2} \cdot 42 \]
\[ S_{21} = 21 \cdot 21 \]
\[ S_{21} = 441 \]

### Final Answer
The sum of the first 21 odd numbers is **441**.

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