What expression represents the sum of the first \(n\) integers?

The expression that represents the sum of the first (n) integers is (\frac{n(n + 1)}{2}). This formula derives from the observation that the sum can be represented as follows:

When adding the first (n) integers together (1, 2, 3, ... up to (n)), a common method to find the total is to rearrange the numbers. If you write the sequence forward and then backwards, pairing the first number with the last, the second with the second to last, and so on, you can see that each of these pairs sums to (n + 1).

For example, if (n = 5), the sums would be:

• 1 + 5 = 6

• 2 + 4 = 6

• The middle number (3, in this case) stands alone.

If (n) is even, there will be (\frac{n}{2}) pairs, and if (n) is odd, there will be (\frac{n - 1}{2}) pairs plus one middle number, which can also be incorporated. In either scenario, the total number of pairs can