1. An infinite sequence is a function whose domain is the set of positive integers. The function values $a_1,a_2,a_3,\dots ,a_n,\dots$ are called the terms of the sequence. The term $a_n$ is the nth term, or general term, of the sequence.

2. Recursive formula

   A sequence may be defined recursively, with the nth term of the sequence defined in relation to previous terms.

3. Factorial notation

   $$n!=n(n-1)\cdots 3\cdot 2\cdot 1\text{ and }0!=1$$

4. Summation notation

   $${\displaystyle \sum _{i=1}^n{a}_i=a_1+a_2+a_3+\cdots +a_n}$$

5. Series

   Given an infinite sequence $a_1,a_2,a_3,\dots ,a_n,\dots$, the sum ${\displaystyle \sum _{i=1}^n{a}_i=a_1+a_2+a_3+\cdots }$ is called a series and the sum ${\displaystyle \sum _{i=1}^n{a}_i}$ is called the nth partial sum of the series.

1. A sequence is an arithmetic sequence if each term after the first differs from the preceding term by a constant. The constant difference d between the consecutive terms is called the common difference. We have $d=a_n-a_{n-1}$ for all $n\ge 2$.

2. The nth term $a_n$ of the arithmetic sequence $a_1,a_2,a_3,\dots ,a_n,\dots$ is given by $a_n=a_1+(n-1)d,n\ge 1$ where d is the common difference.

3. The sum $S_n$ of the first n terms of the arithmetic sequence $a_1,a_2,\dots ,a_n,\dots$ is given by the formula

   $$S_n=n\left({\displaystyle {\displaystyle \frac{a_1+a_n}{2}}}\right)=n\left({\displaystyle {\displaystyle \frac{2a_1+(n-1)d}{2}}}\right).$$

1. A sequence is a geometric sequence if each term after the first is a constant multiple of the preceding term. The constant ratio r between the consecutive terms is called the common ratio. We have ${\displaystyle \frac{a_{n+1}}{a_n}}=r,n\ge 1$.

2. The nth term ${\mathit{a}}_{\mathit{n}}$ of the geometric sequence $a_1,a_2,\dots ,a_n,\dots$ is given by $a_n=a_1{r}^{n-1},n\ge 1$, where r is the common ratio.

3. The sum $S_n$ of the first n term of the geometric sequence $a_1,a_2,\dots ,a_n,\dots$ with common ratio $r\ne 1$ is given by the formula

   $$S_n={\displaystyle {\displaystyle \frac{a_1(1-r^n)}{1-r}}}.$$

   The sum S of an infinite geometric sequence is given by

   $$S={\displaystyle \sum _{i=1}^{\infty }{a}_1{r}^{i-1}={\displaystyle {\displaystyle \frac{a_1}{1-r}}}\text{ if }|r|<1.}$$

4. An annuity is a sequence of equal payments made at equal time intervals. Suppose $P is the payment made at the end of each of n compounding periods year and i is the annual interest rate. Then the value A of the annuity after t year is

   $$A=P{\displaystyle {\displaystyle \frac{{(1+{\displaystyle {\displaystyle \frac{i}{n}}})}^{nt}-1}{{\displaystyle {\displaystyle \frac{i}{n}}}}}}.$$

5. If $\left|r\right|<1$, the infinite geometric series $a_1+a_{1}r+a_1{r}^2+\cdots +a_1{r}^{n-1}+\cdots$ has the sum $S={\displaystyle \frac{a_1}{1-r}}.$ When $\left|r\right|\ge 1,$ the series does not have a finite sum.

The statement $P_n$ is true for all positive integers n if the following properties hold:

1. $P_1$ is true.

2. If $P_k$ is a true statement, then $P_{k+1}$ is a true statement.

1. Binomial coefficient: $\left(\begin{array}{l}n\\ j\end{array}\right)={\displaystyle {\displaystyle \frac{n!}{j!(n-j)!}}},0\le j\le n$

   $$\left(\begin{array}{l}n\\ 0\end{array}\right)=1=\left(\begin{array}{l}n\\ n\end{array}\right)$$

2. Binomial Theorem:

   $$\begin{array}{rcl}{(x+y)}^n& =& \left(\begin{array}{l}n\\ 0\end{array}\right)x^n+\left(\begin{array}{l}n\\ 1\end{array}\right)x^{n-1}y+\left(\begin{array}{l}n\\ 2\end{array}\right)x^{n-2}{y}^2+\cdots \\ & & +\left(\begin{array}{l}n\\ j\end{array}\right)x^{n-j}{y}^j+\cdots +\left(\begin{array}{l}n\\ n\end{array}\right)y^n\\ & =& {\displaystyle \sum _{j=0}^n\left(\begin{array}{l}n\\ j\end{array}\right)x^{n-j}{y}^j}\end{array}$$

3. The term containing the factor $x^r$ in the expansion of ${(x+y)}^n$ is

   $$\left(\begin{array}{c}n\\ n-r\end{array}\right)x^r{y}^{n-r}.$$

1. Fundamental Counting Principle: If a first choice can be made in p different ways, a second choice in q different ways, a third choice in r different ways, and so on, then the sequence of choices can be made in $p\cdot q\cdot r\dots$ different ways.

2. A permutation is an arrangement of n distinct objects in a fixed order in which no object is used more than once. The order in an arrangement is important.

3. Permutation formula: The number of permutations of n distinct objects taken r at a time is

   $$P(n,r)=n(n-1)(n-2)\dots (n-r+1)={\displaystyle {\displaystyle \frac{n!}{(n-r)!}}}$$.

4. Combination formula: When r objects are chosen from n distinct objects without regard to order, we call the set of r objects a combination of n objects taken r at a time. The number of combinations of n distinct objects taken r at a time is denoted by C(n, r), where

   $$C(n,r)={\displaystyle {\displaystyle \frac{n!}{(n-r)!r!}}}.$$

5. Distinguishable permutations: The number of permutations of n objects of which $n_1$ are of one kind, $n_2$ are of a second kind, . . . , and $n_k$ are of a kth kind is

   $${\displaystyle {\displaystyle \frac{n!}{n_1!n_2!\dots n_k{!}^{}}}},$$

   where $n_1+n_2+\cdots +n_k=n$.

1. Experiment: Any process that terminates in one or more outcomes (results) is an experiment.

2. Sample space: The set of all possible outcomes of an ­experiment is called the sample space of the experiment.

3. Event: An event is any subset of a sample space.

4. Equally likely events: Outcomes of an experiment are said to be equality likely if no outcome should result more often than any other outcome when the experiment is performed repeatedly.

5. Probability: If all of the outcomes in a sample space S are equally likely, the probability of an event E, denoted P(E), is defined by

   $$P(E)={\displaystyle {\displaystyle \frac{n(E)}{n(S)}}},$$

   where n(E) is the number of outcomes in E and n(S) is the total number of outcomes in S.

   1. The Additive Rule: If E and F are events in a sample space S, then

      $$P(E\text{ or }F)=P(E\cup F)=P(E)+P(F)-P(E\cap F).$$

   2. If $E\cap F=\varnothing$, E and F are called mutually exclusive events. For mutually exclusive events E and F, $P(E\cup F)=P(E)+P(F)$.

   3. $P(E^{\prime })=P(\text{not }E)=1-P(E);$ the event $E^{\prime }$ is called the complement of the event E.