### Key Concepts

### 9.1 Sequences and Their Notations

- A sequence is a list of numbers, called terms, written in a specific order.
- Explicit formulas define each term of a sequence using the position of the term. See Example 1, Example 2, and Example 3.
- An explicit formula for the $n\text{th}$ term of a sequence can be written by analyzing the pattern of several terms. See Example 4.
- Recursive formulas define each term of a sequence using previous terms.
- Recursive formulas must state the initial term, or terms, of a sequence.
- A set of terms can be written by using a recursive formula. See Example 5 and Example 6.
- A factorial is a mathematical operation that can be defined recursively.
- The factorial of $n$ is the product of all integers from 1 to $n$ See Example 7.

### 9.2 Arithmetic Sequences

- An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
- The constant between two consecutive terms is called the common difference.
- The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See Example 1.
- The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See Example 2 and Example 3.
- A recursive formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{n-1}+d,n\ge 2.$ See Example 4.
- As with any recursive formula, the initial term of the sequence must be given.
- An explicit formula for an arithmetic sequence with common difference $d$ is given by ${a}_{n}={a}_{1}+d(n-1).$ See Example 5.
- An explicit formula can be used to find the number of terms in a sequence. See Example 6.
- In application problems, we sometimes alter the explicit formula slightly to ${a}_{n}={a}_{0}+dn.$ See Example 7.

### 9.3 Geometric Sequences

- A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
- The constant ratio between two consecutive terms is called the common ratio.
- The common ratio can be found by dividing any term in the sequence by the previous term. See Example 1.
- The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See Example 2 and Example 4.
- A recursive formula for a geometric sequence with common ratio $r$ is given by ${a}_{n}=r{a}_{n\u20131}$ for $n\ge 2$ .
- As with any recursive formula, the initial term of the sequence must be given. See Example 3.
- An explicit formula for a geometric sequence with common ratio $r$ is given by ${a}_{n}={a}_{1}{r}^{n\u20131}.$ See Example 5.
- In application problems, we sometimes alter the explicit formula slightly to ${a}_{n}={a}_{0}{r}^{n}.$ See Example 6.

### 9.4 Series and Their Notations

- The sum of the terms in a sequence is called a series.
- A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum. See Example 1.
- The sum of the terms in an arithmetic sequence is called an arithmetic series.
- The sum of the first $n$ terms of an arithmetic series can be found using a formula. See Example 2 and Example 3.
- The sum of the terms in a geometric sequence is called a geometric series.
- The sum of the first $n$ terms of a geometric series can be found using a formula. See Example 4 and Example 5.
- The sum of an infinite series exists if the series is geometric with $\mathrm{\u20131}<r<1.$
- If the sum of an infinite series exists, it can be found using a formula. See Example 6, Example 7, and Example 8.
- An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See Example 9.

### 9.5 Counting Principles

- If one event can occur in $m$ ways and a second event with no common outcomes can occur in $n$ ways, then the first or second event can occur in $m+n$ ways. See Example 1.
- If one event can occur in $m$ ways and a second event can occur in $n$ ways after the first event has occurred, then the two events can occur in $m\times n$ ways. See Example 2.
- A permutation is an ordering of $n$ objects.
- If we have a set of $n$ objects and we want to choose $r$ objects from the set in order, we write $P(n,r).$
- Permutation problems can be solved using the Multiplication Principle or the formula for $P(n,r).$ See Example 3 and Example 4.
- A selection of objects where the order does not matter is a combination.
- Given $n$ distinct objects, the number of ways to select $r$ objects from the set is $\text{C}(n,r)$ and can be found using a formula. See Example 5.
- A set containing $n$ distinct objects has ${2}^{n}$ subsets. See Example 6.
- For counting problems involving non-distinct objects, we need to divide to avoid counting duplicate permutations. See Example 7.

### 9.6 Binomial Theorem

- $\left(\begin{array}{c}n\\ r\end{array}\right)$ is called a binomial coefficient and is equal to $C(n,r).$ See Example 1.
- The Binomial Theorem allows us to expand binomials without multiplying. See Example 2.
- We can find a given term of a binomial expansion without fully expanding the binomial. See Example 3.

### 9.7 Probability

- Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
- The probabilities in a probability model must sum to 1. See Example 1.
- When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See Example 2.
- To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See Example 3.
- To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See Example 4.
- The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See Example 5.
- In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See Example 6.