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\title{Cheat Sheet 1: Sigma Notation}
\author{Simon Harris}
\date{\today}
\maketitle
\section{Removing a Factor}
A simple adjustment which can help reach a situation where \textbf{2} can be applied:
$$
\sum_{i=1}^{n}{3i} = 3\sum_{i=1}^{n}{i}
$$
\section{Finite Arithmetic Sequences}
The sum of a finite arithmetic sequence $1 + 2 + ... + n$ can be written in sigma notation as $\sum_{i=1}^{n}{i}$, but that can alternatively be represented as $\frac{1}{2}n(n + 1)$. So:
$$
\sum_{i=1}^{n}{i} = \frac{1}{2}n(n + 1)
$$
\section{Finite Geometric Sequences}
$$
\sum_{i=0}^{n}{r^i} = \frac{1-r^{n+1}}{1 - r} = \frac{r^{n+1}-1}{r-1}
$$
The second representation is derived by multiplying the first by -1. The former is more convenient where $-1 \le r \le 1$ and the latter for other values of $r$. Combined with \textbf{1}, we can see that:
$$
\sum_{i=0}^{n}{ar^i} = a\left(\frac{1-r^{n+1}}{1 - r}\right) = a\left(\frac{r^{n+1}-1}{r-1}\right)
$$
\section{Another Type of Sequence}
For a sequence $ x_i (i = 1, 2, ...)$ where $a$ and $b$ are constants:
$$
\sum_{i=1}^{n}{a + bx_i} = an + b\sum_{i=1}^{n}{x_i}
$$
Which can be further generalised for sequences beginning at $m$:
$$
\sum_{i=m}^{n}{a + bx_i} = a(n -m + 1) + b\sum_{i=m}^{n}{x_i}
$$
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