More on Sequences

Depending on the sequence pattern, you may sometimes find a handy formula for its sums: with such a formula, each time you are given a number $n$, simply plug the given $n$ into the formula, and you'll quickly find the sum.

Let's consider the sequence $S = \langle1, 2, 3, \dots, n\rangle$ for some fixed integer $n$. It turns out that $$\sum_{i=1}^n i = 1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}.$$

The sum above has a special name: an arithmetic series, and the underlying sequence $S$ is an arithmetic sequence. In this kind of sequence, the difference between each two adjacent terms is the same (in the case of $S$, it is $3 - 2 = 2 - 1 = 1$.)

How was it discovered? A legend/rumor tells that when Carl Friedrich Gauss, an 18th century mathematician, was in 3rd grade, he quickly computed the sum $1 + 2 + \dots + 10$ by noticing that $(10 + 1) + $$(9 + 2) + $$(8 + 3) + $$(7 + 4) + $$(6 + 5) = $$11\cdot5 $$= \frac{10 \cdot 11}{2} $$= 55$. We will prove that the formula is correct later on.