📈College Algebra Unit 13 – Sequences, Probability, and Counting

Key Concepts and Definitions

• Sequence: an ordered list of numbers, typically following a specific pattern or rule
• Term: each number in a sequence, usually denoted as $a_n$ where $n$ represents the position of the term
• Arithmetic sequence: a sequence where the difference between any two consecutive terms is constant (common difference)
• Geometric sequence: a sequence where the ratio between any two consecutive terms is constant (common ratio)
• Probability: a measure of the likelihood that an event will occur, expressed as a number between 0 and 1
• Sample space: the set of all possible outcomes of an experiment or event
• Event: a subset of the sample space, representing a specific outcome or set of outcomes
• Permutation: an arrangement of objects in a specific order, where the order matters
• Combination: a selection of objects from a larger set, where the order does not matter

Sequence Basics

• Sequences can be finite (having a specific number of terms) or infinite (continuing indefinitely)
• The first term of a sequence is denoted as $a_1$, the second term as $a_2$, and so on
• To find the $n$th term of a sequence, a formula or rule is often used, which relates the position $n$ to the value of the term
• The sum of a sequence is the result of adding all the terms together
  • For a finite sequence, the sum can be calculated by adding each term individually
  • For an infinite sequence, the sum may or may not exist, depending on the convergence of the sequence
• Sequences can be increasing (each term is greater than the previous one), decreasing (each term is less than the previous one), or neither
• Sequences can also be monotonic (always increasing or always decreasing) or non-monotonic (changing between increasing and decreasing)

Types of Sequences

• Arithmetic sequences have a constant difference between consecutive terms, denoted as $d$
  • The general term formula for an arithmetic sequence is $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term and $d$ is the common difference
  • Example: 2, 5, 8, 11, 14, ... (common difference of 3)
• Geometric sequences have a constant ratio between consecutive terms, denoted as $r$
  • The general term formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio
  • Example: 3, 6, 12, 24, 48, ... (common ratio of 2)
• Recursive sequences are defined by a recurrence relation, where each term is expressed in terms of the previous term(s)
  • Example: the Fibonacci sequence, where each term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, ...
• Other types of sequences include quadratic sequences, exponential sequences, and logarithmic sequences, each with their own specific patterns and formulas

Probability Fundamentals

• Probability is a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event
• The probability of an event $A$ is denoted as $P(A)$ and is calculated by dividing the number of favorable outcomes by the total number of possible outcomes
• The complement of an event $A$, denoted as $A'$ or $\bar{A}$, represents the event "not $A$"
  • The probability of the complement is given by $P(A') = 1 - P(A)$
• Two events are mutually exclusive if they cannot occur simultaneously
  • The probability of the union of two mutually exclusive events $A$ and $B$ is given by $P(A \cup B) = P(A) + P(B)$
• Two events are independent if the occurrence of one does not affect the probability of the other
  • The probability of the intersection of two independent events $A$ and $B$ is given by $P(A \cap B) = P(A) \cdot P(B)$
• Conditional probability is the probability of an event occurring given that another event has already occurred
  • The conditional probability of event $B$ given event $A$ is denoted as $P(B|A)$ and is calculated by dividing $P(A \cap B)$ by $P(A)$

Counting Principles

• The Multiplication Principle states that if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \cdot n$ ways to do both
• Permutations are used to count the number of ways to arrange $r$ objects from a set of $n$ objects, where the order matters
  • The number of permutations is denoted as $P(n,r)$ or $nPr$ and is calculated using the formula $P(n,r) = \frac{n!}{(n-r)!}$
• Combinations are used to count the number of ways to select $r$ objects from a set of $n$ objects, where the order does not matter
  • The number of combinations is denoted as $C(n,r)$, $nCr$, or $\binom{n}{r}$ and is calculated using the formula $C(n,r) = \frac{n!}{r!(n-r)!}$
• The Binomial Theorem is used to expand $(a+b)^n$ and can be written using combinations as $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

Applications in Real-World Scenarios

• Sequences can be used to model various real-world situations, such as population growth, financial investments, and depreciation
  • Example: compound interest can be modeled using a geometric sequence, where the common ratio represents the growth factor
• Probability is widely used in fields such as insurance, finance, and weather forecasting to make informed decisions based on the likelihood of events
  • Example: insurance companies use probability to determine premiums based on the likelihood of claims
• Counting principles are applied in various fields, including computer science, cryptography, and logistics
  • Example: the number of possible PIN codes for a 4-digit system can be calculated using the Multiplication Principle (10 choices for each digit, so $10^4 = 10,000$ possible codes)
• Combinations and permutations are used in areas such as genetics, chemistry, and game theory
  • Example: the number of possible hands in a card game can be calculated using combinations

Common Mistakes and How to Avoid Them

• Confusing arithmetic and geometric sequences
  • Pay attention to whether the difference or the ratio between consecutive terms is constant
• Misinterpreting probability values
  • Remember that probability is always between 0 and 1, and a higher probability indicates a more likely event
• Misapplying the Multiplication Principle
  • Ensure that the events or choices are independent before multiplying their counts
• Confusing permutations and combinations
  • Permutations are used when the order matters, while combinations are used when the order does not matter
• Forgetting to consider the complement when solving probability problems
  • If asked to find the probability of an event not occurring, calculate the probability of the event occurring and subtract it from 1
• Misusing the Binomial Theorem
  • Make sure to use the correct powers for $a$ and $b$ and to include the appropriate combination coefficient for each term

Practice Problems and Solutions

1. Find the 10th term of the arithmetic sequence: 3, 8, 13, 18, ...

   • Solution: $a_{10} = a_1 + (n-1)d = 3 + (10-1)(5) = 3 + 45 = 48$

2. Find the sum of the first 6 terms of the geometric sequence: 2, 6, 18, 54, ...

   • Solution: $S_6 = \frac{a_1(1-r^6)}{1-r} = \frac{2(1-3^6)}{1-3} = \frac{2(1-729)}{-2} = 728$

3. A fair six-sided die is rolled. What is the probability of rolling an even number?

   • Solution: $P(\text{even}) = \frac{3}{6} = \frac{1}{2}$ (favorable outcomes: 2, 4, 6; total outcomes: 1, 2, 3, 4, 5, 6)

4. How many different 3-letter arrangements can be made using the letters A, B, C, D, and E, if each letter can only be used once?

   • Solution: $P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 60$

5. In a group of 10 people, how many different committees of 4 can be formed?

   • Solution: $C(10,4) = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = 210$