Arithmetic and Geometric Sequences Study Guide

Objectives and Core Learning Goals

• Determine the next term in various numeric sequences.
• Recognize and distinguish between arithmetic sequences and geometric sequences.
• Determine the common difference or common ratio for a specific sequence.
• Graph arithmetic and geometric sequences on a coordinate plane.
• Recognize the graphical behavior (increasing, decreasing, alternating) of sequences.
• Sort sequences that are represented graphically.
• Generate terms of a sequence when provided in function form using a recursive process.

Key Definitions and Fundamental Concepts

• Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is a constant. In other words, a constant is added to each term to produce the next term.
• Common Difference: The constant value added to each term in an arithmetic sequence. It is represented by the variable $d$. It can be positive or negative.
• Geometric Sequence: A sequence of numbers in which the ratio between any two consecutive terms is a constant. This means each term is multiplied by a constant to determine the next term.
• Common Ratio: The constant value (integer or fraction) by which each term in a geometric sequence is multiplied to obtain the next term. It is represented by the variable $r$.
• Sequence Representations: Sequences provide a relationship between term numbers (the position) and term values. All sequences appear as a set of discrete points when graphed and represent functions.

Arithmetic Sequences: In-Depth Study

Characterizing Arithmetic Sequences

• If the same positive number is added to each term, the common difference ($d$) is positive, and the sequence increases.
• If the same negative number is added (which is equivalent to subtracting a positive number), the common difference ($d$) is negative, and the sequence decreases.

Worked Example: Recursive Process

Consider a sequence generated by the formula $a_n = a_{n-1} + (-2)$, where $a_1 = 11$ and $n$ is a whole number greater than 1.

• $a_n$ represents the $n_{th}$ term of the sequence.
• $a_{n-1}$ represents the term immediately before the $n_{th}$ term.
• To find the second term ($a_2$): $a_2 = a_1 + (-2) = 11 + (-2) = 9$.
• To find the third term ($a_3$): $a_3 = a_2 + (-2) = 9 + (-2) = 7$.
• To find the fourth term ($a_4$): $a_4 = a_3 + (-2) = 7 + (-2) = 5$.
• The resulting sequence is $11, 9, 7, 5, \dots$ and the common difference $d$ is $-2$.

Function Form

The sequence can also be written using function notation. For the example where $a_1 = 11$ and the pattern is to subtract 2:

• $f(1) = 11$
• $f(n) = f(n-1) + (-2)$ for $n > 1$

Variations and Effects of Change

If a sequence starts with the same value (11) but is generated by $a_n = a_{n-1} + 4$:

• The first four terms are $11, 15, 19, 23$.
• The common difference $d$ becomes 4.
• The sequence increases by 4 each step instead of decreasing by 2.

Geometric Sequences: In-Depth Study

Characterizing Geometric Sequences

• The common ratio ($r$) can be an integer or a fraction.
• If $r > 1$, the terms increase in magnitude.
• If $0 < r < 1$, the terms decrease in magnitude.
• If $r < 0$, the terms will alternate between positive and negative values.

Worked Example: Recursive Multiplication

Consider a sequence defined by $g_n = g_{n-1} \times 2$, where $g_1 = 1$ and $n$ is a whole number greater than 1.

• The second term: $g_2 = g_1 \times 2 = 1 \times 2 = 2$.
• The third term: $g_3 = g_2 \times 2 = 2 \times 2 = 4$.
• The fourth term: $g_4 = g_3 \times 2 = 4 \times 2 = 8$.
• The resulting sequence is $1, 2, 4, 8, \dots$ and the common ratio $r$ is 2.

Function Form

The above sequence can be expressed as:

• $f(1) = 1$
• $f(n) = 2 \times f(n-1)$ for $n > 1$

Exploration of Ratio Changes

Using the same starting value of 1:

• If the rule is $a_n = a_{n-1} \times 3$, the terms are $1, 3, 9, 27$. The sequence increases more rapidly than with a ratio of 2.
• If the common ratio is $\frac{1}{3}$, the terms are $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}$. This sequence decreases.
• If the sequence is generated by $g_n = 1 \times (-2)^{n-1}$, the first four terms are $1, -2, 4, -8$. This sequence alternates signs.

Analysis of Sequence Cards A-H

Arithmetic Sequences

• Sequence B: $4, \frac{7}{4}, -\frac{1}{2}, -\frac{11}{4}, -5, -\frac{29}{4}, \dots$
  • Rule: Subtract $\frac{9}{4}$.
  • Common difference: $d = -\frac{9}{4}$.
• Sequence D: $-20, -16, -12, -8, -4, 0, 4, \dots$
  • Rule: Add 4.
  • Common difference: $d = 4$.
• Sequence H: $1473.2, 1452.7, 1432.2, 1411.7, 1391.2, 1370.7, \dots$
  • Rule: Subtract 20.5.
  • Common difference: $d = -20.5$.

Geometric Sequences

• Sequence A: $-2, -6, -18, -54, -162, -486, \dots$
  • Rule: Multiply by 3.
  • Common ratio: $r = 3$.
• Sequence E: $-5, -\frac{5}{2}, -\frac{5}{4}, -\frac{5}{8}, -\frac{5}{16}, -\frac{5}{32}, \dots$
  • Rule: Multiply by $\frac{1}{2}$.
  • Common ratio: $r = \frac{1}{2}$.
• Sequence G: $-16, 4, -1, \frac{1}{4}, -\frac{1}{16}, \frac{1}{64}, \dots$
  • Rule: Divide by $-4$ (Multiply by $-\frac{1}{4}$).
  • Common ratio: $r = -\frac{1}{4}$.

Sequences that are Neither Arithmetic nor Geometric

• Sequence C: $1, -2, 3, -4, 5, -6, \dots$
  • Rationale: The values are consecutive numbers, but every other number is negative. There is no constant difference or constant ratio.
• Sequence F: $86, 85, 83, 80, 76, 71, 65, \dots$
  • Rationale: The subtraction amount increases by 1 each time (subtract 1, then 2, then 3, then 4, etc.). Since the difference is not constant, it is neither.

Graphical Behavior and Data

General Properties of Sequence Graphs

• Discrete Points: All sequence graphs consist of discrete points (dots) because the domain is restricted to term numbers (integers).
• Function Verification: Graphs of sequences pass the vertical line test, confirming they are functions.
• Y-Intercept: Sequence graphs typically do not have a y-intercept because sequences start at term number 1 (or sometimes 0), not an intersection with the y-axis in the standard discrete display.

Linear vs. Curved Behavior

• Arithmetic Graphs: The points of an arithmetic sequence lie on a line.
  • Positive $d$ leads to an increasing graph.
  • Negative $d$ leads to a decreasing graph.
• Geometric Graphs: The points of a geometric sequence do not lie on a line; they form a curve.
  • If $r > 1$, the graph increases.
  • If $0 < r < 1$, the graph decreases.
  • If $r < 0$, the graph alternates (zig-zags) between increasing and decreasing between consecutive points.

Graph and Sequence Matching

• Graph 1: Matches Sequence B.
• Graph 2: Matches Sequence A.
• Graph 3: Matches Sequence D.
• Graph 4: Matches Sequence H.
• Graph 5: Matches Sequence E.
• Graph 6: Matches Sequence G.

Questions & Discussion

Common Ratios and Division Consider the sequence $270, 90, 30, 10, \dots$. Jorge says he can determine each term by multiplying by $\frac{1}{3}$, so the common ratio is $\frac{1}{3}$. Jaylen says he determines each term by dividing by 3. While both find the correct next term, only Jorge is technically correct regarding the definition of a common ratio. The common ratio represents the constant by which each term is multiplied, not divided. Therefore, while dividing by 3 is the same as multiplying by $\frac{1}{3}$, the common ratio is formally defined as $r = \frac{1}{3}$.

Ambiguity with Limited Terms If provided only with the first two terms of a sequence, such as $3, 6, \dots$, multiple interpretations are valid:

• Mariana assumes it is arithmetic with $d = 3$, creating the sequence $3, 6, 9, 12, \dots$
• Ashley assumes it is geometric with $r = 2$, creating the sequence $3, 6, 12, 24, \dots$
• The conclusion is that both are correct given only two terms; more information is needed to identify the exact type of sequence.

The Case of Repeating Constants Consider the sequence $2, 2, 2, 2, 2, \dots$. This sequence can be categorized in three ways:

1. Arithmetic: Where the common difference $d = 0$.
2. Geometric: Where the common ratio $r = 1$.
3. Neither: Simply a repeating value of 2.

Generating Sequences from Specifications

1. Specification: First term is 0; common difference is $-6$.
   • Sequence: $0, -6, -12, -18, -24$
   • Type: Arithmetic
2. Specification: First term is $-3$; common ratio is $-\frac{1}{4}$.
   • Sequence: $-3, \frac{3}{4}, -\frac{3}{16}, \frac{3}{64}, -\frac{3}{256}$
   • Type: Geometric