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The image you've provided contains a mathematical problem that asks to determine the sum of the numbers from 1 to 100. Here's a step-by-step solution.

### Problem Restatement:
You are asked to calculate the sum \( S = 1 + 2 + 3 + \ldots + 100 \).

### Explanation:
This is a classic arithmetic series where the first term \( a_1 = 1 \) and the last term \( a_n = 100 \). The sum \( S \) of an arithmetic series can be calculated using the formula:

\[
S = \frac{n}{2} \times (a_1 + a_n)
\]

where:
- \( n \) is the number of terms,
- \( a_1 \) is the first term,
- \( a_n \) is the last term.

### Step-by-Step Calculation:
1. **Determine \( n \)**:  
   Since the numbers are from 1 to 100, \( n = 100 \).

2. **Calculate the sum**:  
   \[
   S = \frac{100}{2} \times (1 + 100)
   \]
   \[
   S = 50 \times 101 = 5050
   \]

### Conclusion:
The sum \( S \) of the numbers from 1 to 100 is \( 5050 \).

Would you like further details or have any other questions?

Here are 5 related questions:
1. How would you calculate the sum of numbers from 1 to 200?
2. What is the formula for the sum of an arithmetic sequence with a common difference greater than 1?
3. How can you derive the formula for the sum of an arithmetic series?
4. What is the sum of the first 50 even numbers?
5. How can the sum of the first 100 odd numbers be calculated?

**Tip:** Knowing how to sum arithmetic series is a useful tool for quickly calculating large sums without adding each individual term.

Explore how to calculate the sum of numbers from 1 to 100 using the arithmetic series formula. Step-by-step solution provided, covering concepts in arithmetic series and number theory. Suitable for Grades 7-9 students and anyone interested in basic number theory and arithmetic sums.

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