Textbook Question
If you toss a fair coin six times, what is the probability of getting all heads?
617
views
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 55
The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = n2 + 5
Verified step by step guidance1
Identify the general term of the sequence: \(a_n = n^{2} + 5\).
Recall that an arithmetic sequence has a constant difference between consecutive terms, called the common difference \(d\), where \(d = a_{n+1} - a_n\).
Calculate the difference between consecutive terms: \(a_{n+1} - a_n = ((n+1)^2 + 5) - (n^2 + 5)\).
Simplify the expression for the difference: expand \((n+1)^2\) to get \(n^2 + 2n + 1\), then subtract \(n^2 + 5\) to find the difference.
Check if the difference is constant (independent of \(n\)). If it is, the sequence is arithmetic with common difference \(d\). If not, check if the ratio \(\frac{a_{n+1}}{a_n}\) is constant to determine if the sequence is geometric. If neither is constant, the sequence is neither arithmetic nor geometric.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3.
Recommended video:
Guided course
5:17
Arithmetic Sequences - General Formula
Geometric Sequence
A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. For example, in 3, 6, 12, 24, the common ratio is 2.
Recommended video:
Guided course
4:18Geometric Sequences - Recursive Formula
General Term of a Sequence
The general term, denoted a_n, defines the nth term of a sequence as a function of n. Understanding this formula helps determine the pattern of the sequence and whether it fits arithmetic or geometric rules.
Recommended video:
Guided course
4:45Geometric Sequences - General Formula
Related Practice
Textbook Question
Find the middle term in the expansion of (3/x + x/3)10
592
views
Textbook Question
Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?
699
views
Textbook Question
In Exercises 51–56, the general term of a sequence is given. Determine whether the sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. an = 2n
877
views
Textbook Question
In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. 5+7+9+11+⋯+ 31
1618
views
Textbook Question
Find f(x + h) − f(x)/h and simplify. f(x) = x4+7
577
views