What will be the sum of odd numbers from 1 to 50 ?

Options

A. 625
B. 650
C. 667
D. 670

Answer

To find the sum of odd numbers from 1 to 50, we can follow these steps:

### Step 1: Identify the Odd Numbers
The odd numbers from 1 to 50 are:
1, 3, 5, 7, 9, ..., 49.

### Step 2: Determine the Number of Terms
The sequence of odd numbers can be expressed as an arithmetic progression (AP) where:
- The first term \(a = 1\)
- The last term \(l = 49\)
- The common difference \(d = 2\)

To find the number of terms \(n\), we can use the formula for the nth term of an AP:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 49 = 1 + (n - 1) \cdot 2 \]
Subtracting 1 from both sides:
\[ 48 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 24 = n - 1 \]
Adding 1 to both sides:
\[ n = 25 \]
So, there are 25 odd numbers from 1 to 50.

### Step 3: Calculate the Sum of the Odd Numbers
The sum \(S_n\) of the first \(n\) terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S_{25} = \frac{25}{2} \cdot (1 + 49) \]
Calculating inside the parentheses:
\[ S_{25} = \frac{25}{2} \cdot 50 \]
Now, simplifying:
\[ S_{25} = 25 \cdot 25 = 625 \]

### Final Answer
The sum of odd numbers from 1 to 50 is **625**.

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