Textbook Question
Find the middle term in the expansion of (3/x + x/3)10
Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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.
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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.
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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.
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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.
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Related Practice
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?
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Textbook Question
Let {an} = - 5, 10, - 20, 40, ..., {bn} = 10, - 5, - 20, - 35, ..., {cn} = - 2, 1, - 1/2, 1/4 Find a10 + b10.
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Textbook Question
Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2
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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
Textbook Question
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58. Find the difference between the sum of the first 14 terms of {bn} and the sum of the first 14 terms of {an}.
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