Book cover for College Algebra

College Algebra

Michael Sullivan

ISBN #9780321979476

10th Edition

5,609 Questions

Group icon
320,854 Students Helped

Homework Questions

Right arrow
Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers different ways to define and work with sequences. It begins by introducing sequences through explicit formulas which allow writing several terms and graphing them. Next, it explores recursively defined sequences (such as factorial and Fibonacci) where each term depends on previous values. The section also introduces summation notation to succinctly represent the sum of sequence terms and provides formulas for common sequences, including some properties for arithmetic sequences. Overall, students learn to express, evaluate, and sum sequences with both explicit and recursive definitions.

Learning Objectives

1

Write the first several terms of a sequence when provided with an explicit formula.

2

Generate terms of a sequence defined recursively and recognize patterns such as factorial and Fibonacci numbers.

3

Apply and interpret summation notation to concisely represent the sum of sequences.

4

Determine the sum of a sequence using given formulas and properties, including arithmetic sequences.

Key Concepts

CONCEPT

DEFINITION

Sequence

A function whose domain is the set of positive integers; its outputs (terms) are listed in order.

nth Term (General Term)

A formula, often denoted aₙ, that gives the value of each term in the sequence based on its position n.

Recursive Formula

A definition of a sequence that specifies one or more initial terms and a recurrence relation to obtain subsequent terms.

Factorial (n!)

For an integer n ≥ 0, n! is defined as n × (n-1) × … × 2 × 1 with 0! defined as 1.

Summation Notation

A compact form using the Greek letter Sigma (Σ) to denote the sum of terms defined by an index variable over a specified range.

Arithmetic Sequence

A sequence in which the difference between successive terms is constant; this difference is called the common difference (d).

Example Problems

Example 1

For the function $f(x)=\frac{x-1}{x},$ find $f(2)$ and $f(3) .$ $(\mathrm{pp} .202-205)$.

Example 2

True or False A function is a relation between two sets $D$ and $R$ so that each element $x$ in the first set $D$ is related to exactly one element $y$ in the second set $R .(\mathrm{pp} .199-202)$

Example 3

A(n) _____ is a function whose domain is the set of positive integers.

Example 4

True or False The notation $a_{5}$ represents the fifth term of a sequence.

Example 5

True or False If $n \geq 2$ is an integer, then $$ n !=n(n-1) \cdots 3 \cdot 2 \cdot 1 $$
Scroll left
Scroll right

Step-by-Step Explanations

QUESTION

Given aₙ = (n - 1)/n for n ≥ 1, write out the first six terms.

STEP-BY-STEP ANSWER:

Step 1: Substitute n = 1 into the formula: a₁ = (1 - 1)/1 = 0.
Step 2: Substitute n = 2: a₂ = (2 - 1)/2 = 1/2.
Step 3: Substitute n = 3: a₃ = (3 - 1)/3 = 2/3.
Step 4: Substitute n = 4: a₄ = (4 - 1)/4 = 3/4.
Step 5: Substitute n = 5: a₅ = (5 - 1)/5 = 4/5.
Step 6: Substitute n = 6: a₆ = (6 - 1)/6 = 5/6.
Final Answer: The first six terms are 0, 1/2, 2/3, 3/4, 4/5, and 5/6.

Writing the First Several Terms of a Sequence

QUESTION

Given the recursive sequence defined by s₁ = 1 and sₙ = n · sₙ₋₁ for n ≥ 2, find the first five terms.

STEP-BY-STEP ANSWER:

Step 1: The first term is provided: s₁ = 1.
Step 2: Compute s₂: s₂ = 2 · s₁ = 2 · 1 = 2.
Step 3: Compute s₃: s₃ = 3 · s₂ = 3 · 2 = 6.
Step 4: Compute s₄: s₄ = 4 · s₃ = 4 · 6 = 24.
Step 5: Compute s₅: s₅ = 5 · s₄ = 5 · 24 = 120.
Final Answer: The first five terms are 1, 2, 6, 24, and 120 (i.e., the factorial sequence).

Writing Terms of a Sequence Defined by a Recursive Formula

QUESTION

Express the sum 1/1 + 1/2 + 1/3 + ... + 1/n using summation notation.

STEP-BY-STEP ANSWER:

Step 1: Recognize each term has the form 1/k, where k starts at 1 and ends at n.
Step 2: Write the summation using Sigma notation: Σₖ₌₁ⁿ (1/k).
Final Answer: The sum can be concisely written as Σₖ₌₁ⁿ (1/k).

Using Summation Notation

QUESTION

Find the sum of the sequence defined by aₖ = k², for k = 1 to n, and write out what the summation notation represents.

STEP-BY-STEP ANSWER:

Step 1: Write the sum in summation notation: Σₖ₌₁ⁿ k².
Step 2: Use the formula for the sum of squares: Σₖ₌₁ⁿ k² = n(n + 1)(2n + 1)/6.
Step 3: Verify for a small value (e.g., n = 3): 1² + 2² + 3² = 1 + 4 + 9 = 14, and the formula gives 3·4·7/6 = 84/6 = 14.
Final Answer: The sum of the squares from k = 1 to n is n(n + 1)(2n + 1)/6.

Finding the Sum of a Sequence

Scroll left
Scroll right

Common Mistakes

  • Assuming that the nth term formula represents the entire sequence without understanding its domain (n must be a positive integer).
  • Confusing recursive formulas with explicit formulas, leading to miscalculation of terms.
  • Incorrectly applying summation notation, such as mismatching the index variable or the summing limits.
  • Forgetting to adjust the index when the summation does not start at 1.
  • Overlooking the importance of the common difference in an arithmetic sequence.