Miscellaneous Sequences

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A list of random formula's and notations that are useful to know.

Authors: Jaden Park, Leo Zhang

Notations

\sum_{r = 1}^{n}{r}

is the same as saying

1 + 2 + 3 + 4 + \ldots + n

In english, it means "The sum of $r$ from $r = 1$ to $n.$ " If it were different, replace $r$ or the variables with the new variable or value.

$\sum_{r = 1}^{n}{(a_r \pm b_r)} = \sum_{r = 1}^{n}{a_r} \pm \sum_{r = 1}^{n}{b_r}$
$\sum_{r = 1}^{n}{k a_r} = k \sum_{r = 1}^{n} a_r$
$\sum_{r = 1}^{n}{k + k + k + k + \ldots + n} = nk$ where $k$ is constant.
$\sum_{r = 1}^{n}{r} = 1 + 2 + 3 + 4 + 5 + \ldots + n = \frac{n(n + 1)}{2}$
$\sum_{r = 1}^{n}{r^{2}} = 1^2 + 2^2 + 3^2 + \ldots + n^2 = \frac{n(n + 1)(2n + 1)}{6}$
$\sum_{r = 1}^{n}{r^{3}} = 1^3 + 2^3 + 3^3 + \ldots + n^3 = \frac{n^{2}(n + 1)^{2}}{4}$

Recursion

Floor Functions