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If you toss a fair coin six times, what is the probability of getting all heads?
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
9장, 문제 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
검증된 단계별 안내1
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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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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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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.
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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.
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4:45Geometric Sequences - General Formula
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