How many terms of the series `1+2+3…………..` add upto 5050

Options

A. 50
B. 51
C. 100
D. 101

Answer

To find how many terms of the series \(1 + 2 + 3 + \ldots\) add up to 5050, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Series**:
The series given is the sum of the first \(n\) natural numbers:
\[ S_n = 1 + 2 + 3 + \ldots + n \]

2. **Use the Formula for the Sum of the First \(n\) Natural Numbers**:
The formula for the sum of the first \(n\) natural numbers is:
\[ S_n = \frac{n(n + 1)}{2} \]
We know that this sum equals 5050:
\[ \frac{n(n + 1)}{2} = 5050 \]

3. **Multiply Both Sides by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ n(n + 1) = 5050 \times 2 \]
\[ n(n + 1) = 10100 \]

4. **Set Up the Quadratic Equation**:
Rearranging gives us a standard quadratic equation:
\[ n^2 + n - 10100 = 0 \]

5. **Solve the Quadratic Equation**:
We can use the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 1\), and \(c = -10100\):
\[ n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-10100)}}{2 \cdot 1} \]
\[ n = \frac{-1 \pm \sqrt{1 + 40400}}{2} \]
\[ n = \frac{-1 \pm \sqrt{40401}}{2} \]
\[ n = \frac{-1 \pm 201}{2} \]

6. **Calculate the Values**:
This gives us two potential solutions:
\[ n = \frac{200}{2} = 100 \quad \text{(valid since \(n\) must be positive)} \]
\[ n = \frac{-202}{2} = -101 \quad \text{(not valid)} \]

7. **Conclusion**:
Therefore, the number of terms \(n\) that add up to 5050 is:
\[ n = 100 \]

### Final Answer:
The number of terms of the series that add up to 5050 is **100**.