Chapter Study Guide: Sequences

Learning Objectives

By the end of this study guide, you should be able to:

• Find the formula for the general term of a sequence.
• Calculate the limit of a sequence if it exists.
• Determine the convergence or divergence of a given sequence.

Key Terms & Glossary

• Infinite Sequence: An ordered list of numbers of the form $a_1, a_2, a_3, \dots, a_n, \dots$
• Term: Each individual number in a sequence (e.g., $a_n$ is the $n^{\text{th}}$ term).
• Index Variable: The subscript (usually $n$) indicating the position of a term within the sequence.
• Explicit Formula: A mathematical expression that computes the $n^{\text{th}}$ term $a_n directly as a function of the index n$.
• Recurrence Relation: A mathematical expression defining the $n^{\text{th}}$ term $a_n$ based on one or more preceding terms in the sequence.
• Arithmetic Sequence: A sequence where the difference between any pair of consecutive terms is a constant value.
• Geometric Sequence: A sequence where the ratio of any pair of consecutive terms is a constant value.

The "Big Idea"

Sequences can be fundamentally understood as functions whose domain is restricted to the positive integers. Instead of the continuous curves you studied in differential calculus, sequences represent discrete, ordered lists of values. By finding explicit mathematical patterns (formulas) for these lists, we can analyze their long-term behavior as $n$ approaches infinity. This forms the foundational concept for understanding whether an infinite list of numbers settles on a specific limit (convergence) or fails to do so (divergence), ultimately paving the way for the study of infinite series.

[!NOTE] Although sequences most commonly begin with an index of $n = 1, they can technically start at n = 0$ or any other integer depending on the mathematical context.

Formula / Concept Box

Hierarchical Outline

• 1. Terminology of Sequences
  • Infinite Sequence Definition: Denoted using set notation as $\{a_n\}_{n=1}^{\infty}$, representing a strictly ordered list.
  • Index Variable: The $n$ in $a_n$, anchoring the term to a specific position.
  • Generation Methods: Sequences are built either by an explicit formula (plugging in $n$) or a recurrence relation (bootstrapping from previous terms).
• 2. Graphing Sequences
  • Because a sequence is a function mapped over positive integers, points are plotted as coordinates $(n, a_n)$.
  • The graph forms a set of discrete, disconnected dots.
• 3. Common Classifications
  • Arithmetic Sequences: Maintain a constant difference ($d$).
  • Geometric Sequences: Maintain a constant ratio ($r$).
  • Alternating Sequences: Flip signs between positive and negative, constructed using an oscillating factor like $(-1)^n$.

Visual Anchors

Definition-Example Pairs

• Infinite Sequence
  • Definition: A function whose domain is the set of positive integers, resulting in an ordered, never-ending list.
  • Real-World Example: The sequence of leap years starting from 2000: $\{2000, 2004, 2008, 2012, \dots\}$.
• Arithmetic Sequence
  • Definition: A sequence where each term equals the previous term plus a constant difference $d$.
  • Real-World Example: A taxi meter that starts at a $3 base fare and adds $2 per mile. The cost at mile $n$ forms the sequence: $\{3, 5, 7, 9, \dots\}$.
• Geometric Sequence
  • Definition: A sequence where each term equals the previous term multiplied by a constant ratio $r$.
  • Real-World Example: The population of a bacteria colony that doubles every hour. If it starts at 100, the hourly populations form the sequence: $\{100, 200, 400, 800, \dots\}$.
• Alternating Sequence
  • Definition: A sequence whose terms strictly alternate between positive and negative values.
  • Real-World Example: The net balance sequence of a bank account where a person alternates between depositing $50 and withdrawing $50 every day: $\{50, -50, 50, -50, \dots\}$.

Worked Examples

Example 1: Finding an Explicit Formula for an Alternating Sequence

Problem: Find an explicit formula for the $n^{\text{th}} term of the sequence \{-\frac{2}{5}, \frac{5}{8}, -\frac{8}{11}, \frac{11}{14}, \dots\}$.

Step-by-Step Solution:

1. Handle the alternating sign: The sequence goes negative, positive, negative, positive. Because the odd terms ($n=1, 3) are negative, the sequence must include a factor of (-1)^n$.
2. Analyze the numerators: The absolute values of the numerators are $2, 5, 8, 11. The difference between consecutive terms is 3. This is an arithmetic sequence with a_1=2$and$d=3$. The explicit formula for the numerator is $2 + 3(n-1) = 3n - 1$.
3. Analyze the denominators: The denominators are $5, 8, 11, 14. The constant difference is 3. This is an arithmetic sequence with a_1=5$and$d=3$. The explicit formula for the denominator is $5 + 3(n-1) = 3n + 2$.
4. Combine the components: $a_n = (-1)^n \frac{3n - 1}{3n + 2}$

Example 2: Converting a Recurrence Relation to an Explicit Formula

Problem: A sequence is defined by the recurrence relation $a_1 = 3$ and $a_n = 2a_{n-1}$ for $n \ge 2$. Find its explicit formula.

Step-by-Step Solution:

1. Write out the first few terms to spot the pattern:
   • $a_1 = 3$
   • $a_2 = 2(3) = 6$
   • $a_3 = 2(6) = 12$
   • $a_4 = 2(12) = 24$
2. Identify the sequence type: The ratio of consecutive terms is consistently 2. Therefore, this is a geometric sequence where the common ratio $r = 2 and the starting term a_1 = 3$.
3. Apply the general explicit geometric formula: Substitute the values into $a_n = a_1 \cdot r^{n-1}$.
4. Final Formula: $a_n = 3(2)^{n-1}$

Checkpoint Questions

▶1. What is the fundamental difference between an explicit formula and a recurrence relation?

An explicit formula allows you to calculate the $n^{\text{th}} term directly using only the index position n$ (e.g., $a_n = 2n). A recurrence relation requires you to calculate or know previous terms in order to find the next term (e.g., a_n = a_{n-1} + 2$).

▶2. Is the sequence $a_n = \frac{n}{n+1}$ an arithmetic sequence, a geometric sequence, or neither?

Neither. If we calculate the first few terms: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \dots$

The difference is not constant ($\frac{2}{3} - \frac{1}{2} = \frac{1}{6}$, but $\frac{3}{4} - \frac{2}{3} = \frac{1}{12}$).

The ratio is also not constant ($\frac{2/3}{1/2} = \frac{4}{3}$, but $\frac{3/4}{2/3} = \frac{9}{8}$).

▶3. If a sequence is treated as a mathematical function, how does its graph visually differ from a typical function like $f(x) = x^2$?

Because the domain of a sequence is restricted to discrete positive integers ($n=1, 2, 3, \dots$), its graph consists only of disconnected, individual coordinate points rather than a continuous, smooth curve.

▶4. Write the explicit formula for a geometric sequence where the first term is 5 and the ratio between consecutive terms is 3.

$a_n = 5(3)^{n-1}$