Book cover for Precalculus with Limits

Precalculus with Limits

Ron Larson

ISBN #9781439049099

2nd Edition

7,319 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces sequences and series as fundamental mathematical structures used to model and analyze ordered collections of numbers. It covers writing sequences in subscript notation, employing factorial and summation notation, and calculating series sums. Real-life applications such as compound interest and population growth highlight the practical importance of these concepts, while both explicit and recursive definitions allow versatile representations.

Learning Objectives

1

Write and interpret sequences using subscript notation and appropriate functional expressions.

2

Utilize factorial notation in sequence definitions and calculations.

3

Express sums using summation (sigma) notation and compute series sums.

4

Apply the concepts of sequences and series to model real-life problems such as compound interest and population growth.

5

Distinguish between finite and infinite sequences, as well as explicit and recursive sequence definitions.

Key Concepts

CONCEPT

DEFINITION

Sequence

An ordered list of numbers defined as the values of a function whose domain is a set of integers; sequences can be finite (limited number of terms) or infinite.

Subscript Notation

A notation such as aₙ that identifies the nᵗʰ term in a sequence.

Recursive Sequence

A sequence in which one or more initial terms are given and each subsequent term is defined using one or more previous terms; the Fibonacci sequence is a classic example.

Factorial (n!)

For a positive integer n, n! is the product of all positive integers up to n; by definition, 0! = 1.

Summation (Sigma) Notation

A compact notation using the Greek letter sigma (Σ) to represent the sum of a sequence of terms, with specified lower and upper limits.

Series

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

Example Problems

Example 1

Fill in the blanks. An ________ ________ is a function whose domain is the set of positive integers.

Example 2

Fill in the blanks. The function values $ a_1, a_2, a_3, a_4, \cdots $ are called the ________ of a sequence.

Example 3

Fill in the blanks. A sequence is a ________ sequence if the domain of the function consists only of the first $ n $ positive integers.

Example 4

Fill in the blanks. 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 said to be defined ________.

Example 5

Fill in the blanks. If $ n $ is a positive integer, $ n $ ________ is defined as $ n ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n - 1) \cdot n $.
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Step-by-Step Explanations

QUESTION

Given the sequence defined by aₙ = 3n - 2, write the first four terms.

STEP-BY-STEP ANSWER:

Step 1: Identify the rule: aₙ = 3n - 2.
Step 2: For n = 1, compute a₁ = 3(1) - 2 = 1.
Step 3: For n = 2, compute a₂ = 3(2) - 2 = 4.
Step 4: For n = 3, compute a₃ = 3(3) - 2 = 7.
Step 5: For n = 4, compute a₄ = 3(4) - 2 = 10.
Final Answer: The first four terms are 1, 4, 7, 10.

Writing the Terms of a Sequence

QUESTION

Evaluate 4!.

STEP-BY-STEP ANSWER:

Step 1: Recall the definition: 4! = 1 × 2 × 3 × 4.
Step 2: Multiply sequentially: 1 × 2 = 2, then 2 × 3 = 6, then 6 × 4 = 24.
Final Answer: 4! = 24.

Evaluating a Factorial Expression

QUESTION

Express and compute the sum of the first four terms of the sequence aₙ = 3n - 2 using sigma notation.

STEP-BY-STEP ANSWER:

Step 1: Write the sum in sigma notation: Σₙ₌₁⁴ (3n - 2).
Step 2: Calculate each term as previously computed: 1, 4, 7, 10.
Step 3: Add the terms: 1 + 4 + 7 + 10 = 22.
Final Answer: The sum is 22.

Using Summation Notation

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Common Mistakes

  • Failing to specify the starting index when writing sequences, leading to ambiguous definitions.
  • Miscomputing factorial values, especially confusing 0! as 0 instead of 1.
  • Improperly setting the limits in summation notation, which can lead to errors in series calculations.
  • Assuming that listing a few terms is enough to define a sequence without providing the general nth-term formula.
  • Overlooking the differences between explicit and recursive sequence definitions.