A student of `5^(th)` standard started writing down the counting number as `1,2,3,4,..` and then he added all those numbers and got the result 500 . But when I checkedthe resultI have found that he had missed a number. What is the missing number?

Options

A. 25
B. 32
C. 30
D. 28

Answer

To find the missing number that the student did not include in his sum of counting numbers, we can follow these steps:

### Step 1: Understand the Problem
The student added the counting numbers from 1 to n and got a sum of 500, but he missed one number. We need to find out which number he missed.

### Step 2: Use the Formula for the Sum of Natural Numbers
The formula for the sum of the first n natural numbers is given by:
\[ S = \frac{n(n + 1)}{2} \]
Where \( S \) is the sum and \( n \) is the last number in the counting sequence.

### Step 3: Set Up the Equation
Since the student got a sum of 500, we can set up the equation:
\[ \frac{n(n + 1)}{2} - x = 500 \]
Where \( x \) is the missing number.

### Step 4: Rearranging the Equation
Rearranging the equation gives us:
\[ \frac{n(n + 1)}{2} = 500 + x \]
Multiplying both sides by 2:
\[ n(n + 1) = 1000 + 2x \]

### Step 5: Estimate n
We need to find a suitable \( n \) such that \( n(n + 1) \) is close to 1000. We can try different values for \( n \):
- For \( n = 31 \):
\[ 31 \times 32 = 992 \]
- For \( n = 32 \):
\[ 32 \times 33 = 1056 \]

Since \( 992 < 1000 < 1056 \), we can conclude that \( n \) is likely 31 or 32.

### Step 6: Calculate the Sum for n = 31
Using \( n = 31 \):
\[ \frac{31 \times 32}{2} = 496 \]
This means if the student had written all numbers from 1 to 31, the sum would be 496. Since he got 500, the missing number \( x \) can be calculated as:
\[ 496 + x = 500 \implies x = 500 - 496 = 4 \]

### Step 7: Verify with n = 32
Now let's check for \( n = 32 \):
\[ \frac{32 \times 33}{2} = 528 \]
This means if the student had written all numbers from 1 to 32, the sum would be 528. Since he got 500, the missing number \( x \) can be calculated as:
\[ 528 - x = 500 \implies x = 528 - 500 = 28 \]

### Conclusion
The student could have missed either the number 4 (if he wrote up to 31) or the number 28 (if he wrote up to 32). However, since the problem states he wrote counting numbers and missed one, the more plausible answer is that he missed the number 28, as it is larger and would affect the sum more significantly.

Thus, the missing number is:
\[ \boxed{28} \]