Michael Sullivan
ISBN #9780321716811
9th Edition
5,201 Questions
Homework Questions
This section covers the foundations of sequences, including both explicit and recursive definitions, the use of summation notation, and properties of factorials. The key concepts include how to generate terms, graph sequences, and apply these ideas to real-world problems such as financial calculations, approximation of functions, and natural growth processes such as the Fibonacci sequence. Understanding these principles provides a strong basis for more advanced topics such as induction and the Binomial Theorem.
1
Describe the concept of a sequence, including explicit (nth term) and recursive definitions.
2
Explain how to represent and graph sequences using formulas, table features, and graphing utilities.
3
Apply summation notation to write and simplify the sum of sequences with varied indices and expressions.
4
Understand and compute factorial values and recognize their roles in sequence definitions.
5
Identify and analyze arithmetic sequences and their properties including common differences.
CONCEPT
DEFINITION
Sequence
An ordered list of numbers defined by either an explicit (nth term) formula or a recursive formula.
Nth term (general term)
A formula that directly calculates the value of the term in the nth position of the sequence without reference to previous terms.
Recursive Sequence
A sequence defined by one or more initial terms and a rule that relates each term to one or more of its predecessors.
Factorial (n!)
The product of all positive integers up to n (with 0! defined as 1). It is used in many sequences and summation expressions.
Summation Notation
A concise expression using the Greek letter sigma (Σ) to indicate that a sum is taken over a set of terms indexed by an integer.
Arithmetic Sequence
A sequence in which the difference between consecutive terms is constant (known as the common difference, d).
For the function $f(x)=\frac{x-1}{x},$ find $f(2)$ and $f(3)$.
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 .
A(n)_________is a function whose domain is the set of $A(n)$positive integers.
True or False The notation $a_{5}$ represents the fifth term of a sequence.
If $n \geq 0$ is an integer, then $n !=$ _________when $n=2$.
QUESTION
Given the sequence defined by a_n = 1/n, how do you find the 5th term?
STEP-BY-STEP ANSWER:
Step 1: Identify the formula for the nth term, which is a_n = 1/n. Step 2: Substitute n = 5 into the formula. Step 3: Calculate 1/5. Final Answer: The 5th term is 1/5.
Finding the nth Term of a Sequence
For the recursive sequence defined by s_1 = 1 and s_n = n * s_(n-1), find s_3.
Step 1: Recognize that s_2 = 2 * s_1 = 2 * 1 = 2. Step 2: Then, s_3 = 3 * s_2 = 3 * 2 = 6. Final Answer: The third term, s_3, is 6.
Evaluating a Recursive Sequence
Write the sum Σ (k^2) from k=1 to 4 as a standard addition.
Step 1: Substitute k = 1, 2, 3, 4 into the expression k^2. Step 2: Compute each term: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16. Step 3: Add the computed values. Final Answer: 1 + 4 + 9 + 16 = 30.
Expanding Summation Notation