### Key Concepts

## 12.1 Sequences

**Factorial Notation**

If*n*is a positive integer, then $n!$ is

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

We define $0!$ as 1, so $0!=1$**Summation Notation**

The sum of the first*n*terms of a sequence whose*n*th 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 (***n*th term) of an Arithmetic Sequence

The general term of an arithmetic sequence with first term ${a}_{1}$ and the common difference*d*is

$${a}_{n}={a}_{1}+\left(n-1\right)d$$**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*n*th term is

$${S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})$$

## 12.3 Geometric Sequences and Series

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

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

$${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$$

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*,

$\begin{array}{}\\ \\ \phantom{\rule{1.5em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}\left|r\right|<1,\phantom{\rule{0.2em}{0ex}}\text{the sum is}\hfill \\ \\ \phantom{\rule{5em}{0ex}}{S}_{}=\frac{{a}_{1}}{1-r}\hfill \\ \phantom{\rule{1.5em}{0ex}}\text{We say the series converges.}\hfill \\ \phantom{\rule{1.5em}{0ex}}\text{If}\phantom{\rule{0.2em}{0ex}}\left|r\right|\ge 1,\phantom{\rule{0.2em}{0ex}}\text{the infinite geometric series does not have a sum. We say the series diverges.}\hfill \end{array}$**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

$${A}_{t}=\frac{P\left({\left(1+\frac{r}{n}\right)}^{nt}-1\right)}{\frac{r}{n}}$$

## 12.4 Binomial Theorem

**Patterns in the expansion of**${\left(a+b\right)}^{n}$- 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**$(\begin{array}{c}n\hfill \\ r\hfill \end{array})$**:**A binomial coefficient $\left(\begin{array}{c}n\hfill \\ r\hfill \end{array}\right),$ where*r*and*n*are integers with $0\le r\le n,$ is defined as

$$\left(\begin{array}{c}n\hfill \\ r\hfill \end{array}\right)=\frac{n!}{r!\left(n-r\right)!}$$

We read $\left(\begin{array}{c}n\hfill \\ r\hfill \end{array}\right)$ as “*n*choose*r*” or “*n*taken*r*at a time”.**Properties of Binomial Coefficients**

$\begin{array}{ccccccccc}\hfill \left(\begin{array}{c}n\hfill \\ 1\hfill \end{array}\right)=n\hfill & & & & \hfill \left(\begin{array}{c}n\hfill \\ n\hfill \end{array}\right)=1\hfill & & & & \hfill \left(\begin{array}{c}n\hfill \\ 0\hfill \end{array}\right)=1\hfill \end{array}$**Binomial Theorem:**For any real numbers*a*,*b*, and positive integer*n*,

$${(a+b)}^{n}=\left(\begin{array}{c}n\hfill \\ 0\hfill \end{array}\right){a}^{n}+\left(\begin{array}{c}n\hfill \\ 1\hfill \end{array}\right){a}^{n-1}{b}^{1}+\left(\begin{array}{c}n\hfill \\ 2\hfill \end{array}\right){a}^{n-2}{b}^{2}+\mathrm{...}+\left(\begin{array}{c}n\hfill \\ r\hfill \end{array}\right){a}^{n-r}{b}^{r}+\mathrm{...}+\left(\begin{array}{c}n\hfill \\ n\hfill \end{array}\right){b}^{n}$$