The least value of n such that 1 + 3 + 5 + 7 + ….. N terms `ge` 500 is

Options

A. 18
B. 19
C. 22
D. 23

Answer

To find the least value of \( n \) such that the sum of the first \( n \) odd numbers \( 1 + 3 + 5 + 7 + \ldots \) is greater than or equal to 500, we can follow these steps:

### Step 1: Understand the series
The series \( 1, 3, 5, 7, \ldots \) is an arithmetic progression (AP) where:
- The first term \( a = 1 \)
- The common difference \( d = 2 \)

### Step 2: Sum of the first \( n \) odd numbers
The formula for the sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 2) \]
This simplifies to:
\[ S_n = \frac{n}{2} \times (2 + 2n - 2) = \frac{n}{2} \times 2n = n^2 \]

### Step 3: Set up the inequality
We need to find the smallest \( n \) such that:
\[ n^2 \geq 500 \]

### Step 4: Solve the inequality
Taking the square root of both sides:
\[ n \geq \sqrt{500} \]
Calculating \( \sqrt{500} \):
\[ \sqrt{500} = \sqrt{100 \times 5} = 10\sqrt{5} \approx 10 \times 2.236 = 22.36 \]
Since \( n \) must be a whole number, we round up to the next integer:
\[ n \geq 23 \]

### Step 5: Verify the value of \( n \)
Now, we check if \( n = 23 \) satisfies the condition:
\[ S_{23} = 23^2 = 529 \]
Since \( 529 \geq 500 \), \( n = 23 \) is valid.

### Conclusion
Thus, the least value of \( n \) such that the sum of the first \( n \) odd numbers is greater than or equal to 500 is:
\[ \boxed{23} \]