12.1 Sequences

Factorial Notation
If n is a positive integer, then $n!$ is

$n! = n (n - 1) (n - 2) \dots (3) (2) (1) .$

We define $0!$ as 1, so $0! = 1$
Summation Notation
The sum of the first n terms of a sequence whose nth term $a_{n}$ is written in summation notation as:

$\sum_{i = 1}^{n} a_{i} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + \dots + a_{n}$

The i is the index of summation and the 1 tells us where to start and the n tells us where to end.

12.2 Arithmetic Sequences

General Term (nth term) of an Arithmetic Sequence
The general term of an arithmetic sequence with first term $a_{1}$ and the common difference d is

$a_{n} = a_{1} + (n - 1) d$
Sum of the First n Terms of an Arithmetic Sequence
The sum, $S_{n},$ of the first n terms of an arithmetic sequence, where $a_{1}$ is the first term and $a_{n}$ is the nth term is

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$

General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term $a_{1}$ and the common ratio r is

$a_{n} = a_{1} r^{n - 1}$
Sum of the First n Terms of a Geometric Series: The sum, $S_{n},$ of the n terms of a geometric sequence is

$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

where $a_{1}$ is the first term and r is the common ratio.
Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is $a_{1}$ and common ratio is r and is written

$a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1} + \dots$
Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is $a_{1}$ and common ratio r,
$\begin{matrix} If | r | < 1, the sum is \\ S_{} = \frac{a_{1}}{1 - r} \\ We say the series converges. \\ If | r | \geq 1, the infinite geometric series does not have a sum. We say the series diverges. \end{matrix}$
Value of an Annuity with Interest Compounded $n$ Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

$A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}}$

Patterns in the expansion of (a+b)n(a+b)n
- The number of terms is $n + 1 .$
- The first term is $a^{n}$ and the last term is $b^{n} .$
- The exponents on a decrease by one on each term going left to right.
- The exponents on b increase by one on each term going left to right.
- The sum of the exponents on any term is n.
Pascal’s Triangle
Binomial Coefficient $(\begin{matrix} n \\ r \end{matrix})$ : A binomial coefficient $(\begin{matrix} n \\ r \end{matrix}),$ where r and n are integers with $0 \leq r \leq n,$ is defined as

$(\begin{matrix} n \\ r \end{matrix}) = \frac{n!}{r! (n - r)!}$

We read $(\begin{matrix} n \\ r \end{matrix})$ as “n choose r” or “n taken r at a time”.
Properties of Binomial Coefficients
$\begin{matrix} (\begin{matrix} n \\ 1 \end{matrix}) = n & (\begin{matrix} n \\ n \end{matrix}) = 1 & (\begin{matrix} n \\ 0 \end{matrix}) = 1 \end{matrix}$
Binomial Theorem: For any real numbers a, b, and positive integer n,

${(a + b)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) a^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a^{n - 1} b^{1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{n - 2} b^{2} + ... + (\begin{matrix} n \\ r \end{matrix}) a^{n - r} b^{r} + ... + (\begin{matrix} n \\ n \end{matrix}) b^{n}$