AII.2a 
The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve realworld problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include S and an.
*Standard AII.2 will be assessed in the Functions and Statistics reporting category. (Revised March 2011) 

Essential Knowledge and Skills 
 Distinguish between a sequence and a series.
 Generalize patterns in a sequence using explicit and recursive formulas.
 Use and interpret the notations å, n, nth term, and an.
 Given the formula, find an (the nth term) for an arithmetic or a geometric sequence.
 Given formulas, write the first n terms and find the sum, Sn, of the first n terms of an arithmetic or geometric series.
 Given the formula, find the sum of a convergent infinite series.
 Model realworld situations using sequences and series.

Essential Understandings 
 Sequences and series arise from realworld situations.
 The study of sequences and series is an application of the investigation of patterns.
 A sequence is a function whose domain is the set of natural numbers.
 Sequences can be defined explicitly and recursively.


Vertical Articulation 
 SOL 7.2  describe/represent arithmetic/geometric sequences
using variable expressions

Instructional Materials 
 VDOE ESS Lesson: Arithmetic and Geometric Sequences and Series  Expressions and Operations (PDF)  Exploring sequences and series (Word)

Real World Lessons 

Practice 
 VDOE ESS: TicTacToe Worksheet (pdf)
 VDOE ESS: Applications of Sequences and Series Worksheet (pdf)
 VDOE ESS: Infinite Geometric Seuences Worksheet (pdf)
 VDOE ESS: The Motionless Runner Worksheet (pdf)

SelfAssessments 
 SOL MiniQuiz: AII.2  Series

Skills 
 Mathematics Assessment Project
 A06:Sidewalk Patterns  Apprentice  In Prague some sidewalks are made of small square blocks of stone. The blocks are in different shades to make patterns that are in various sizes. In this task, you will look for rules which let you work out how many blocks of different colors are needed to make different sized patterns. (This task is very similar to Sidewalk Stones, but offers more stepbystep guidance.)
 A20:Multiplying Cells  Apprentice  Students in a science lab observe cells multiplying. Your task is to use their observation to work out how the number of cells increases over time.
 E13:Sidewalk Stones  Expert  This task is very similar to Sidewalk Patterns, but offers less stepbystep guidance.

Videos 
 Video Tutor  Arithmetic Series
 Video Tutor  Geometric Series
 Video Tutor  Infinite Geometric Series

Explore
Learning 
 Arithmetic Sequences  Find the value of individual terms in arithmetic sequences using graphs of the sequences and direct computation. Vary the common difference and examine how the sequences change in response.
 Geometric Sequences  Explore geometric sequences by varying the initial term and the common ratio and examining the graph. Compute specific terms in the sequence using the explicit and recursive formulas.
 Arithmetic and Geometric Sequences  Find the value of individual terms in an arithmetic or geometric sequence using graphs of the sequence and direct computation. Vary the common difference and common ratio and examine how the sequence changes in response.

Websites 
 Notes: Arithmetic Series and Sequences notes
 Notes: Geometric Series and Sequences notes
 Interactive Notes: Geometric Series
 Example: Visualize an Infinite Series

Common Core
Standards 
NRN.2 
ASSE.4 
FIF.3 
FBF.2 
FLE.2 
