Book cover for Algebra and Trigonometry Real Mathematics, Real People

Algebra and Trigonometry Real Mathematics, Real People

Ron Larson

ISBN #9781305071735

7th Edition

6,909 Questions

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234,597 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section on sequences and series introduces various notations and techniques essential for modeling patterns within ordered sets of numbers. Readers learn to write sequences explicitly and recursively, manipulate factorial expressions, employ summation notation to represent series, and determine both finite and infinite sums. Applications such as compound interest and population dynamics illustrate real-world uses of these mathematical concepts.

Learning Objectives

1

Use sequence notation to write the terms of a sequence and identify the nth term.

2

Apply factorial notation and simplify expressions involving factorials.

3

Utilize summation (sigma) notation to represent finite and infinite series and compute their sums.

4

Model and solve real-life problems such as compound interest and population dynamics using sequences and series.

5

Differentiate between algebraic and recursive definitions of sequences, and understand how to work with each.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers defined by a rule. It can be given explicitly by a formula for the nth term (e.g., an = (-1)^n/(2n-1)) or defined recursively.

Series

The sum of the terms of a sequence. A finite series sums a finite number of terms, and an infinite series sums an infinite number of terms.

Factorial (n!)

The product of all positive integers up to n, defined as n! = 1∙2∙3∙...∙n with 0! defined as 1.

Summation Notation (Sigma Notation)

A concise way of writing sums using the Greek letter sigma (∑) to indicate the sum of terms, with specified lower and upper limits.

Partial Sum

The sum of the first n terms of a series, often used to estimate the total sum of an infinite series.

Recursive Sequence

A sequence in which one or more initial terms are given and subsequent terms are defined based on previous term(s).

Example Problems

Example 1

Fill in the blank(s). The function values $a_{1}, a_{2}, a_{3}, a_{4}, \ldots, a_{n}, \ldots$ are called the _____ of a sequence.

Example 2

Fill in the blank(s). If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is defined _____.

Example 3

Fill in the blank(s). For the sum $\sum_{i=1}^{n} a_{i}, i$ is called the _____ of summation, $n$ is the _____ of summation, and 1 is the _____ of summation.

Example 4

Fill in the blank(s). The sum of the terms of a finite or an infinite sequence is called a _____.

Example 5

Which describes an infinite sequence? a finite sequence? (a) The domain consists of the first $n$ positive integers. (b) The domain consists of the set of positive integers.
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Step-by-Step Explanations

QUESTION

Write the first five terms of the sequence given by an = (-1)^n/(2n-1).

STEP-BY-STEP ANSWER:

Step 1: Substitute n = 1 into the formula: a1 = (-1)^1/(2*1-1) = -1/1 = -1.
Step 2: Substitute n = 2: a2 = (-1)^2/(2*2-1) = 1/3.
Step 3: Substitute n = 3: a3 = (-1)^3/(2*3-1) = -1/5.
Step 4: Substitute n = 4: a4 = (-1)^4/(2*4-1) = 1/7.
Step 5: Substitute n = 5: a5 = (-1)^5/(2*5-1) = -1/9.
Final Answer: The first five terms are -1, 1/3, -1/5, 1/7, -1/9.

Writing Terms of a Sequence

QUESTION

Simplify the expression (8!)/(2!∙6!).

STEP-BY-STEP ANSWER:

Step 1: Write out 8! as 8∙7∙6! and note that 6! cancels with the denominator.
Step 2: The expression becomes (8∙7∙6!)/(2!∙6!) = (8∙7)/(2!).
Step 3: Since 2! = 2, simplify to (8∙7)/2 = 56/2 = 28.
Final Answer: 28.

Simplifying Factorial Expressions

QUESTION

Express the sum 3 + 6 + 9 + 12 + 15 using summation notation and find its value.

STEP-BY-STEP ANSWER:

Step 1: Recognize the sequence is arithmetic: Each term increases by 3.
Step 2: Write the nth term as: an = 3n, with n starting at 1.
Step 3: Express the sum as ∑ (n=1 to 5) 3n.
Step 4: Factor out the constant: 3∑ (n=1 to 5) n.
Step 5: Use the formula for the sum of the first 5 natural numbers: (5(5+1))/2 = 15.
Step 6: Multiply by 3: 3 × 15 = 45.
Final Answer: The sum is 45.

Using Summation Notation

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Common Mistakes

  • Confusing sequence notation with summation notation, leading to errors in identifying the range of indices.
  • Misapplying factorial notation by not properly grouping expressions (e.g., mistaking 2n! for (2n)! instead of 2(n!) ).
  • Overlooking the need to include the correct starting index in recursive sequences, which can lead to incorrect term computations.
  • Failing to recognize that listing a few terms does not uniquely define a sequence without the nth term formula.