SS - Modeling with Sequences and Series Module Overview

Modeling with Sequences and Series Module Overview

Introduction

There are many number patterns found in nature. The Fibonacci sequence is used to describe patterns you see on sea shells, pinecones and broccoli. Number patterns exist in business too. Lists of numbers can be modeled by sequences and help us to predict the next term from the previous. When the terms of a sequence is added, a series is formed. Knowing how to analyze these number patterns allows people to make predictions which drive business decisions. In this module, we will learn how to find numbers in a sequence, the sum of a series, and determine if a the limit of a sequence exists.

Essential Questions

What is a sequence?
How do I determine if a sequence is arithmetic or geometric?
How do I write the formula for the nth term of an arithmetic sequence or geometric sequence?
How do I find the sum of the first n terms of an arithmetic series or a geometric series?
How can I determine if a limit of a sequence exists?
How is summation notation used to represent the partial sum of a series?

Key Terms

The following key terms will help you understand the content in this module.

Arithmetic Sequence – A sequence of numbers where the the difference of consecutive terms is constant.

Arithmetic Series – The sum of the first n terms in an arithmetic sequence.

Common Difference – The constant difference between terms of an arithmetic sequence.

Common Ratio – The constant ratio between any two consecutive terms in a geometric sequence.

Explicit Rule – A formula that allows you to find the nth term of a sequence by substituting known values in the expression.

Explicit Formula for an Arithmetic Sequence:	Explicit Formula for a Geometric Sequence:
$a_n=a_1+d(n-1)$ where $a_1$ is the first term and d is the common difference.	$LaTeX: a_n=a_1r^{n-1}$ $a_n=a_1r^{n-1}$ where $a_1$ is the first term and r is the common ratio.

Geometric Sequence – A sequence of numbers where the ratio of every two consecutive terms is constant.

Geometric Series – The sum of the first n terms of a geometric sequence.

Partial Sum – The sum of the first n terms of an infinite series. The partial sum may approach a limiting value.

Recursive Rule – A rule written to describe the relationship between the successive terms in a sequence.

Recursive formula for an Arithmetic Sequence:	Recursive formula for a Geometric Sequence:
$LaTeX: a_n=a_{n-1}+d$ $a_n=a_{n-1}+d$ where d is the common difference.	$LaTeX: a_n=ra_{n-1}$ $a_n=ra_{n-1}$ for $LaTeX: n\ge2$ $n\ge2$ and r is the common ratio.

Sequence – A function whose domain is a set of consecutive integers.

Series – The resulting expression when the terms of a sequence are added together. A series can be finite or infinite.

Summation Notation – A notation to express the sum of a series. Example: $LaTeX: \sum_{i=1}^42i$ $\sum_{i=1}^42i$ represents the sum of this series: 2+4+6+8, with the lower limit is i=1 and the upper limit is 4.

Sigma Notation – Another name for summation notation, using the Greek letter sigma: $LaTeX: \sum^{}$ $\sum^{}$

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