What is the square root of the sum of first 20 odd natural numbers?

Options

A. 10
B. 30
C. 20
D. 40

Answer

To solve the problem of finding the square root of the sum of the first 20 odd natural numbers, we can follow these steps:

### Step 1: Identify the first 20 odd natural numbers
The first 20 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 sequence as an arithmetic progression (AP)
The sequence of odd natural numbers is an arithmetic progression where:
- The first term (a) = 1
- The common difference (d) = 2
- The number of terms (n) = 20

### Step 3: Use the formula for the sum of an arithmetic progression
The formula for the sum (S) of the first n terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 4: Substitute the known values into the formula
Substituting the values into the formula:
- n = 20
- a = 1
- d = 2

The sum becomes:
\[ S_{20} = \frac{20}{2} \times (2 \times 1 + (20 - 1) \times 2) \]

### Step 5: Simplify the expression
Calculating step-by-step:
1. Calculate \( \frac{20}{2} = 10 \)
2. Calculate \( 2 \times 1 = 2 \)
3. Calculate \( (20 - 1) = 19 \)
4. Calculate \( 19 \times 2 = 38 \)
5. Now, combine these: \( 2 + 38 = 40 \)
6. Finally, multiply: \( 10 \times 40 = 400 \)

So, the sum of the first 20 odd natural numbers is \( 400 \).

### Step 6: Find the square root of the sum
Now, we need to find the square root of the sum:
\[ \sqrt{400} = 20 \]

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

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