More on Sequences

In a general case, the sum of an arithmetic sequence of the form $A $$\;= \langle a_1,\;a_2,\;a_3,\;\dots,\;a_n\rangle $$\;= \langle a_1,\;a_1 + d, $$\;a_1 + 2d, $$\;\dots, $$\;a_1 +d\cdot(n-1)\rangle$, where $d$ is the common difference, is

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Let's get back to $S = \langle1, 2, 3, \dots, n\rangle$. If you apply the product notation to $S$, you get $\boldsymbol n$ factorial: $$n! = \prod_{i=1}^n i = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n.$$ Note that $0! = 1$, $1! = 1$, $2! = 2\cdot1 = 2$, etc. It doesn't have a formula, but most calculators have an $n!$ button.