Natural Number

To solve the problem, we need to find how many two-digit numbers are divisible by 7.

1. Define the range of two-digit numbers:
Two-digit numbers range from 10 to 99.

2. Find the smallest two-digit number divisible by 7:
The smallest two-digit number is 10.
Divide 10 by 7:
Next multiple of 7 is:

3. Find the largest two-digit number divisible by 7:
The largest two-digit number is 99.
Divide 99 by 7:
Largest multiple of 7 within range:

4. Count the number of multiples from 14 to 98:
This is an arithmetic sequence:
Use the formula for the nth term of an AP:
, where ,
We solve:

Final Answer:
The correct answer is option (D): 13.

1. Understanding the Sum of First n Natural Numbers:
The sum of the first n natural numbers is a well-known mathematical formula:

2. Evaluating the Given Options:
Let’s quickly go over the choices:

• Option (1): — This gives the sum of first (n − 1) natural numbers.
• Option (2): — This is the correct formula.
• Option (3): — This is a rearranged negative version, not correct.
• Option (4): — This is incorrect and overcomplicates the expression.

3. Example Check (Optional for Validation):
If n = 5, then the sum should be:

Using Option (2):
✔️

Final Answer:
The correct formula is

Step 1: Recall the formula for the sum of the first natural numbers: Step 2: Substitute into the formula: Step 3: Simplify the expression inside the bracket: Step 4: Multiply the numbers: Step 5: Divide the result by 2: Hence, the sum of the first 100 natural numbers is .

The correct option is (D):