Problem Statement:
Evaluate the expression 1 + 2 + 3 + 4 + 5 + ... + 100.

Solution:
To solve this problem without using a formula, we can use a simple approach of adding the numbers one by one until we reach 100.

Approach:
1. Initialize a variable, let's say "sum," to store the cumulative sum of the numbers.
2. Start with the first number, which is 1, and add it to the sum.
3. Move to the next number, which is 2, and add it to the sum.
4. Continue this process until we reach the number 100, adding each number to the sum.
5. Finally, the value of the sum will be the answer to the expression.

Step-by-Step Calculation:
Let's calculate the sum step-by-step to understand the process better.

1. sum = 1 (initialize the sum)
2. sum = 1 + 2 = 3
3. sum = 3 + 3 = 6
4. sum = 6 + 4 = 10
5. sum = 10 + 5 = 15
6. sum = 15 + 6 = 21
7. sum = 21 + 7 = 28
8. sum = 28 + 8 = 36
9. sum = 36 + 9 = 45
10. sum = 45 + 10 = 55

Continuing this process, we will eventually reach:

100. sum = 5,050 + 100 = 5,150

Answer:
The value of the expression 1 + 2 + 3 + 4 + 5 + ... + 100 is 5,150.

Conclusion:
By adding each number one by one until we reach 100, we can find the sum without using a formula. This approach is straightforward and can be applied to any similar expressions.