교과서 질문
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 = n + 5
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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 53
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58. If {an} is a finite sequence whose last term is -83, how many terms does {an} contain?
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Identify the first term and the common difference of the arithmetic sequence from the given points. The first term \(a_1\) corresponds to the value when \(n=1\), which is 4. The common difference \(d\) can be found by subtracting consecutive terms, for example, \(7 - 4\) or \(10 - 7\).
Write the general formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n - 1)d\).
Substitute the known values of \(a_1\) and \(d\) into the formula to express \(a_n\) in terms of \(n\).
Since the last term of the sequence is given as \(-83\), set \(a_n = -83\) and solve the equation \(-83 = 4 + (n - 1)d\) for \(n\).
Solve the resulting equation for \(n\) to find the number of terms in the sequence.
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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 each term after the first is found by adding a constant difference to the previous term. This difference is called the common difference. For example, the sequence 4, 7, 10 has a common difference of 3.
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가이드 코스
5:17
Arithmetic Sequences - General Formula
General Formula for the nth Term
The nth term of an arithmetic sequence can be found using the formula a_n = a_1 + (n - 1)d, where a_1 is the first term, d is the common difference, and n is the term number. This formula helps find any term in the sequence without listing all terms.
추천 영상:
가이드 코스
5:17Arithmetic Sequences - General Formula
Solving for the Number of Terms
To find how many terms a finite arithmetic sequence contains when the last term is known, set the nth term formula equal to the last term and solve for n. This involves algebraic manipulation to isolate n, giving the total number of terms in the sequence.
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4:47The Number e
관련 실천
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Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?
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교과서 질문
In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)
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교과서 질문
Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58.
Find a14+b12.
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교과서 질문
Express each sum using summation notation. Use as the lower limit of summation and for the index of summation.
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교과서 질문
Find f(x + h) − f(x)/h and simplify. f(x) = x4+7
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