Sequences, Series, and Probability: Study Notes for College Algebra

Section 8.1: Sequences and Summation Notation

Introduction to Sequences

A sequence is an ordered list of numbers, often defined by a formula for the nth term or by a recurrence relation. Sequences are foundational in mathematics and appear in various real-world contexts, such as population growth, financial calculations, and natural phenomena like the Fibonacci sequence.

• General Term (nth term): The formula that defines the value of each term in the sequence as a function of its position n.

• Example: For $a_n = 2n + 5$, the first four terms are 7, 9, 11, 13.

Recursion Formulas

A recursive formula defines each term of a sequence using previous terms.

• Example: $a_1 = 3$, $a_n = 2a_{n-1} + 5$ for $n \geq 2$ yields 3, 11, 27, 59.

Factorial Notation

The factorial of a positive integer n, denoted $n!$, is the product of all positive integers up to n. Factorials are used in sequence formulas and combinatorics.

• Example: $a_n = \frac{20}{(n+1)!}$ gives the first four terms: 10, 3.33, 1.67, 0.83 (rounded).

Summation Notation

Summation notation (sigma notation) is a concise way to represent the sum of a sequence of terms.

• Example: $\sum_{k=3}^{5} (2^k + 3)$ expands to $(8+3) + (16+3) + (32+3) = 11 + 19 + 35 = 65$.

Section 8.2: Arithmetic Sequences

Definition and Common Difference

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant value, called the common difference (d), to the previous term.

• General term: $a_n = a_1 + (n-1)d$

• Example: For $a_1 = 100$, $d = -30$, the first six terms are 100, 70, 40, 10, -20, -50.

Sum of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence is given by:

• $S_n = \frac{n}{2}(a_1 + a_n)$

• Example: For the sequence 3, 6, 9, ..., the sum of the first 15 terms is $S_{15} = \frac{15}{2}(3 + 45) = 360$.

Section 8.3: Geometric Sequences and Series

Definition and Common Ratio

A geometric sequence is a sequence where each term after the first is obtained by multiplying the previous term by a constant called the common ratio (r).

• General term: $a_n = a_1 r^{n-1}$

• Example: For $a_1 = 12$, $r = \frac{1}{2}$, the first six terms are 12, 6, 3, 1.5, 0.75, 0.375.

Sum of a Geometric Sequence

• Finite sum: $S_n = a_1 \frac{1 - r^n}{1 - r}$, for $r \neq 1$

• Infinite sum (|r| < 1): $S = \frac{a_1}{1 - r}$

• Example: For $a_1 = 2$, $r = 3$, the sum of the first 9 terms is $S_9 = 2 \frac{1 - 3^9}{1 - 3} = 9842$.

Applications: Annuities and Salary Growth

• Annuity formula: $A = P \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}}$

• Example: Saving $100/month$ at 9.5% compounded monthly for 35 years yields $A \approx 333,946$.

Section 8.4: Mathematical Induction

Principle of Mathematical Induction

Mathematical induction is a proof technique used to establish that a statement holds for all positive integers.

1. Base Case: Prove the statement for $n = 1$.

2. Inductive Step: Assume true for $n = k$, then prove for $n = k + 1$.

• Example: Prove $2 + 2^2 + 2^3 + ... + 2^n = 2^{n+1} - 2$ for all $n \geq 1$.

Section 8.5: The Binomial Theorem

Binomial Coefficients

The binomial coefficient $\binom{n}{r}$ counts the number of ways to choose r objects from n without regard to order.

• $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

• Example: $\binom{6}{3} = 20$

Binomial Expansion

The Binomial Theorem gives the expansion of $(a + b)^n$:

• $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

• Example: $(x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1$

Finding a Particular Term

• The $(r+1)$th term in $(a + b)^n$ is $\binom{n}{r} a^{n-r} b^r$

• Example: The fifth term in $(2x + y)^9$ is $\binom{9}{4} (2x)^5 y^4 = 126 \times 32x^5 y^4 = 4032x^5 y^4$

Section 8.6: Counting Principles, Permutations, and Combinations

Fundamental Counting Principle

If an event can occur in m ways and another independent event in n ways, the total number of ways both can occur is $m \times n$.

• Example: 3 sizes, 4 crusts, 6 toppings: $3 \times 4 \times 6 = 72$ pizzas.

Permutations

A permutation is an arrangement of objects where order matters.

• $P(n, r) = \frac{n!}{(n - r)!}$

• Example: 7 people, 4 offices: $P(7, 4) = 840$ ways.

Combinations

A combination is a selection of objects where order does not matter.

• $C(n, r) = \frac{n!}{r!(n - r)!}$

• Example: 10 physicians, 4 selected: $C(10, 4) = 210$ ways.

Section 8.7: Probability

Empirical Probability

Empirical probability is based on observed data.

• $P(E) = \frac{\text{number of times E occurs}}{\text{total number of trials}}$

• Example: Probability of a positive mammogram: $P = \frac{7664}{100,000} = 0.077$

Theoretical Probability

Theoretical probability is based on the possible outcomes in a sample space.

• $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

• Example: Probability of rolling a 5 or 6 on a die: $P = \frac{2}{6} = \frac{1}{3}$

Probability of Complements

• $P(\text{not } E) = 1 - P(E)$

• Example: Probability a card is not a king: $P = 1 - \frac{4}{52} = \frac{12}{13}$

Probability of Unions and Intersections

• Mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$

• Not mutually exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Independent events: $P(A \text{ and } B) = P(A) \times P(B)$

Table: Example of Empirical Probability

	Breast Cancer	No Breast Cancer
Positive Mammogram	720	6944
Negative Mammogram	80	92,256

Main purpose: To compute probabilities based on observed frequencies.

Summary Table: Key Formulas

Concept	Formula
Arithmetic Sequence (nth term)	$a_n = a_1 + (n-1)d$
Arithmetic Series (sum of n terms)	$S_n = \frac{n}{2}(a_1 + a_n)$
Geometric Sequence (nth term)	$a_n = a_1 r^{n-1}$
Geometric Series (sum of n terms)	$S_n = a_1 \frac{1 - r^n}{1 - r}$
Geometric Series (infinite, |r| < 1)	$S = \frac{a_1}{1 - r}$
Binomial Coefficient	$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
Permutation	$P(n, r) = \frac{n!}{(n - r)!}$
Combination	$C(n, r) = \frac{n!}{r!(n - r)!}$
Empirical Probability	$P(E) = \frac{\text{number of times E occurs}}{\text{total number of trials}}$
Theoretical Probability	$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Additional info: These notes include expanded explanations, examples, and formulas to ensure a self-contained, comprehensive study guide for exam preparation in college algebra.