Arithmetic and Geometric Sequences

Lesson Overview and Learning Objectives

The study of arithmetic and geometric sequences involves recognizing numeric patterns, determining subsequent terms, and identifying the mathematical rules that govern sequence progression.

• Sequence Definitions: Proficiency in defining arithmetic sequences, common differences, geometric sequences, and common ratios.
• Graphical Representation: Identification of the graphical behavior of sequences, specifically distinguishing between linear and non-linear discrete plots.
• Function Notation: Usage of recursive processes to identify terms of sequences given in function form ($f(n)$).
• Categorization: Sorting sequences into groups (Arithmetic, Geometric, or Neither) based on common characteristics and providing rationales.

Essential Ideas of Sequences

• Sequence Basics: A sequence is a relationship between term numbers and term values. All sequence graphs are represented by a set of discrete points.
• Arithmetic Sequences: A sequence where the difference between any two consecutive terms is a constant.
  • This constant is the common difference, represented by the variable $d$.
  • It can be viewed as adding a constant (positive or negative) to each term to produce the next.
• Geometric Sequences: A sequence where the ratio between any two consecutive terms is a constant.
  • This constant is the common ratio, represented by the variable $r$.
  • It is determined by multiplying each term by the constant to produce the next.
• The Concept of Consecutive Terms: In mathematics, terms are consecutive when they follow one another in order without gaps, such as the days of the week or numbers on a number line.

Graphical Behavior of Sequences

Graphs provide a visual method to identify trends and predict future terms in a sequence.

• Discrete Nature: Graphs of sequences do not have a $y$-intercept and consist of discrete points because term numbers are typically restricted to whole numbers or integers ($n=1, 2, 3, \dots$).
• Arithmetic Sequence Graphs:
  • The points of an arithmetic sequence always lie on a line (linear).
  • If the common difference ($d$) is positive, the graph is increasing.
  • If the common difference ($d$) is negative, the graph is decreasing.
• Geometric Sequence Graphs:
  • The points of a geometric sequence do not lie on a line; they form a curve.
  • If the common ratio ($r$) is greater than $1$, the graph is increasing.
  • If the common ratio ($r$) is between $0$ and $1$, the graph is decreasing.
  • If the common ratio ($r$) is less than $0$, the graph alternates between increasing and decreasing between consecutive points (oscillating).

Analysis of Specific Numeric Sequences

Based on the analysis of sequences $A$ through $H$, the following determinations were made regarding their rules and classifications:

• Sequence A: $-2, -6, -18, -54, -162, -486, \dots$

  • Rule: Multiply by $3$.
  • Classification: Geometric sequence with a common ratio $r = 3$.
  • Graph: Graph 2 (Decreasing curve starting at $(1, -2)$).

• Sequence B: $4, \frac{7}{4}, -\frac{1}{2}, -\frac{11}{4}, -5, -\frac{29}{4}, \dots$

  • Rule: Subtract $\frac{9}{4}$ (or add $-\frac{9}{4}$).
  • Classification: Arithmetic sequence with a common difference $d = -\frac{9}{4}$.
  • Graph: Graph 1 (Decreasing line).

• Sequence C: $1, -2, 3, -4, 5, -6, \dots$

  • Rule: Consecutive numbers with alternating signs.
  • Classification: Neither arithmetic nor geometric.

• Sequence D: $-20, -16, -12, -8, -4, 0, 4, \dots$

  • Rule: Add $4$.
  • Classification: Arithmetic sequence with a common difference $d = 4$.
  • Graph: Graph 3 (Increasing line).

• Sequence E: $-5, -\frac{5}{2}, -\frac{5}{4}, -\frac{5}{8}, -\frac{5}{16}, -\frac{5}{32}, \dots$

  • Rule: Multiply by $\frac{1}{2}$.
  • Classification: Geometric sequence with a common ratio $r = \frac{1}{2}$.
  • Graph: Graph 5 (Increasing curve below the $x$-axis).

• Sequence F: $86, 85, 83, 80, 76, 71, 65, \dots$

  • Rule: Subtract $1$, then $2$, then $3$, continuing the pattern.
  • Classification: Neither arithmetic nor geometric.

• Sequence G: $-16, 4, -1, \frac{1}{4}, -\frac{1}{16}, \frac{1}{64}, \dots$

  • Rule: Divide by $-4$ (which is multiplying by $-\frac{1}{4}$).
  • Classification: Geometric sequence with a common ratio $r = -\frac{1}{4}$.
  • Graph: Graph 6 (Alternating points above and below the $x$-axis).

• Sequence H: $1473.2, 1452.7, 1432.2, 1411.7, 1391.2, 1370.7, 1350.2, \dots$

  • Rule: Subtract $20.5$.
  • Classification: Arithmetic sequence with a common difference $d = -20.5$.
  • Graph: Graph 4 (Decreasing line).

Worked Examples and Function Forms

Arithmetic Recursive Example

Consider a sequence where $a_n = a_{n-1} + (-2)$, with the first term $a_1 = 11$.

• $a_1 = 11$
• $a_2 = a_1 + (-2) = 11 - 2 = 9$
• $a_3 = a_2 + (-2) = 9 - 2 = 7$
• $a_4 = a_3 + (-2) = 7 - 2 = 5$
• Resulting Sequence: $11, 9, 7, 5, \dots$
• Function Form: $f(1) = 11$; $f(n) = f(n-1) - 2$

Geometric Recursive Example

Consider a sequence where $g_n = g_{n-1} \times 2$, with the first term $g_1 = 1$.

• $g_1 = 1$
• $g_2 = g_1 \times 2 = 1 \times 2 = 2$
• $g_3 = g_2 \times 2 = 2 \times 2 = 4$
• $g_4 = g_3 \times 2 = 4 \times 2 = 8$
• Resulting Sequence: $1, 2, 4, 8, \dots$
• Function Form: $f(1) = 1$; $f(n) = 2f(n-1)$

The Ambiguity of Two Terms

Given only the first two terms of a sequence, such as $3, 6, \dots$, the type of sequence cannot be definitively determined:

• If assumed arithmetic ($d=3$), the sequence is $3, 6, 9, 12, \dots$
• If assumed geometric ($r=2$), the sequence is $3, 6, 12, 24, \dots$

Questions & Discussion

Question: When you subtract the same number each time to determine the next value in a sequence, is the sequence arithmetic? Response: Yes. Subtracting a positive number is mathematically equivalent to adding a negative number. Because the difference between consecutive terms remains a negative constant, it fits the definition of an arithmetic sequence.

Question: In the sequence $270, 90, 30, 10, \dots$, Jorge says the common ratio is $\frac{1}{3}$ and Jaylen says he determines terms by dividing by $3$. Who is correct? Response: Both utilize the correct logic to find the next term, but Jorge is more technically accurate regarding the definition of a "common ratio." A common ratio must be expressed as the multiplier. While dividing by $3$ works, the common ratio is specifically $\frac{1}{3}$.

Question: How does the common ratio determine if a geometric sequence increases or decreases? Response: If the terms are positive and the ratio is greater than $1$, the sequence increases. If the ratio is between $0$ and $1$, the sequence decreases toward zero. If the ratio is negative, the values alternate between positive and negative, regardless of whether the absolute values are increasing or decreasing.

Question: Can a sequence be both arithmetic and geometric? Response: Consider the sequence $2, 2, 2, 2, 2, \dots$. It can be seen as arithmetic (adding $0$) or geometric (multiplying by $1$). It could also simply be described as a repeating sequence of the same term.

Formulas for Sequence Representations

Arithmetic Summary

• Sequence B: Recursive: $a_n = a_{n-1} - \frac{9}{4}$; Explicit: $a_n = -\frac{9}{4}n + 6.25$
• Sequence D: Recursive: $a_n = a_{n-1} + 4$; Explicit: $a_n = 4n - 24$
• Sequence H: Recursive: $a_n = a_{n-1} - 20.5$; Explicit: $a_n = -20.5n + 1493.7$

Geometric Summary

• Sequence A: Recursive: $a_n = a_{n-1} \times 3$; Explicit: $a_n = -2 \times 3^{n-1}$
• Sequence E: Recursive: $a_n = a_{n-1} \times \frac{1}{2}$; Explicit: $a_n = (-5) \times (\frac{1}{2})^{n-1}$
• Sequence G: Recursive: $a_n = a_{n-1} \times (-\frac{1}{4})$; Explicit: $a_n = -16 \times (-\frac{1}{4})^{n-1}$