## The Formula for the Sum of Numbers from 1 to 100

There's a handy formula to calculate the sum of consecutive numbers from 1 to any given number, including 100. This formula is especially useful for large numbers where adding them manually would be tedious and time-consuming.

### The Formula

The formula to calculate the sum of numbers from 1 to n is:

**Sum = n * (n + 1) / 2**

Where **n** is the last number in the sequence.

### Applying the Formula to 1 to 100

To find the sum of numbers from 1 to 100, we substitute **n = 100** into the formula:

**Sum = 100 * (100 + 1) / 2**

**Sum = 100 * 101 / 2**

**Sum = 5050**

Therefore, the sum of numbers from 1 to 100 is **5050**.

### Why This Formula Works

This formula derives from the concept of pairing numbers in the sequence. If we arrange the numbers from 1 to 100 in ascending and descending order, we notice:

1 + 100 = 101 2 + 99 = 101 3 + 98 = 101 ... 50 + 51 = 101

We have 50 pairs, each summing up to 101. Therefore, the total sum is 50 * 101 = 5050. This principle applies to any consecutive number sequence.

### Conclusion

The formula **n * (n + 1) / 2** provides an efficient way to calculate the sum of consecutive numbers, eliminating the need for manual addition, especially for larger sequences. Understanding the underlying logic of pairing numbers helps visualize why this formula works so effectively.