Field Guide Worksheet Sequence Cipher
STUDENT FIELD GUIDE: ARITHMETIC & GEOMETRIC PATTERNS
Subject: Algebra 2
Date:
Agent Name:
Phase 1: Pattern Play
Analyze the sequences below. Identify the next two numbers and describe the pattern in your own words.
Sequence Alpha:
2, 4, 6, 8, ___, ___
The Rule:
Sequence Beta:
2, 4, 8, 16, ___, ___
The Rule:
Phase 2: Decoding Key
Arithmetic Sequence
Definition
Pattern based on: ___________________
Common Difference (\(d\))
Explicit Formula
\(a_n = \)
Geometric Sequence
Definition
Pattern based on: ___________________
Common Ratio (\(r\))
Explicit Formula
\(a_n = \)
Phase 3: Formula Hunters
Classify each sequence as Arithmetic (A) or Geometric (G) . Identify the first term (\(a_1\)) and the common difference (\(d\)) or ratio (\(r\)), then write the explicit formula.
Sequence Type \(a_1\) \(d\) or \(r\) Explicit Formula (\(a_n\)) 2, 5, 8, 11, ... 3, 6, 12, 24, ... 100, 90, 80, 70, ... 1, 4, 16, 64, ... 10, 5, 2.5, 1.25, ... -5, 0, 5, 10, ... 81, 27, 9, 3, ... 1/2, 1, 3/2, 2, ... 7, 7.7, 8.4, 9.1, ... 2, -4, 8, -16, ...
Bonus Challenge:
Find the geometric mean and arithmetic mean between 4 and 64.
ARITHMETIC MEAN:
GEOMETRIC MEAN:
Sequence Cipher Slides Algebra 2: Sequences
SEQUENCE CIPHER
Decoding the patterns of Arithmetic and Geometric progressions.
Pattern Play
Sequence Alpha
2, 4, 6, 8, ...
What comes next? What's the rule?
Sequence Beta
2, 4, 8, 16, ...
What comes next? What's the rule?
Investigation Video
Watch: 0:00 - 7:05
Embedded media
Focus Questions:
The Two Paths
Arithmetic
Based on Addition/Subtraction
Uses Common Difference (\(d\))
\(a_n = a_1 + (n-1)d\)
Geometric
Based on Multiplication/Division
Uses Common Ratio (\(r\))
\(a_n = a_1 \cdot r^{n-1}\)
Finding the Middle
Arithmetic Mean
The "Average"
\(\frac{a + b}{2}\)
Geometric Mean
The "Square Root of Product"
\(\sqrt{a \cdot b}\)
The mean of two non-adjacent terms in a sequence always gives you the term exactly in the middle!
Formula Hunters
Open your Field Guide. Your mission is to analyze 10 distinct sequences, classify their behavior, and extract their explicit formulas.
Classify (A or G)
Find \(a_1\)
Find \(d\) or \(r\)
Write \(a_n\)
Mission Debrief
Complete the final sequence cipher before you leave:
Target Sequence:
5, 15, 45, ...
What is the explicit formula for the \(n\)th term?
Final Code Exit Ticket FINAL CODE
Sequence Cipher Exit Ticket
Mission ID: Algebra 2
Agent Name
Mission Date
Given the following sequence:
5, 15, 45, ...
1 Identify the first term (\(a_1\)) and the pattern behavior (\(d\) or \(r\)).
First Term (\(a_1\)):
Value of \(d\) or \(r\):
2 Write the explicit formula for the \(n\)th term.
\(a_n = \)
3 Use your formula to find the 6th term.
Cipher Clearance Required for Departure
Cipher Key Teacher Guide Cipher Key
TEACHER FACILITATION GUIDE & ANSWER KEY
Algebra 2 | Sequences
Lesson Objective
Students will be able to distinguish between arithmetic and geometric sequences, calculate means, and derive explicit formulas for the \(n\)th term.
Key Vocabulary
Arithmetic: Common difference (\(d\))
Geometric: Common ratio (\(r\))
Explicit: Directly finds term \(n\)
Mean: Middle value in sequence
Pacing
Pattern Play 5 min
Video Segment 10 min
Formula Hunters 20 min
Exit Ticket 10 min
Phase 3: Formula Hunters Answer Key
Sequence Type \(a_1\) \(d\) or \(r\) Explicit Formula (\(a_n\)) 2, 5, 8, 11... A 2 \(d = 3\) \(a_n = 2 + (n-1)3\) or \(3n-1\) 3, 6, 12, 24... G 3 \(r = 2\) \(a_n = 3(2)^{n-1}\) 100, 90, 80, 70... A 100 \(d = -10\) \(a_n = 100 + (n-1)(-10)\) 1, 4, 16, 64... G 1 \(r = 4\) \(a_n = 1(4)^{n-1}\) 10, 5, 2.5... G 10 \(r = 0.5\) \(a_n = 10(0.5)^{n-1}\) -5, 0, 5, 10... A -5 \(d = 5\) \(a_n = -5 + (n-1)5\) or \(5n-10\) 81, 27, 9... G 81 \(r = 1/3\) \(a_n = 81(1/3)^{n-1}\) 1/2, 1, 3/2, 2... A 1/2 \(d = 1/2\) \(a_n = 1/2 + (n-1)(1/2)\) or \(n/2\) 7, 7.7, 8.4... A 7 \(d = 0.7\) \(a_n = 7 + (n-1)0.7\) 2, -4, 8, -16... G 2 \(r = -2\) \(a_n = 2(-2)^{n-1}\)
Exit Ticket Key
Sequence: 5, 15, 45...
