Textbook Question
Evaluate each expression.
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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 23
Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. 1, 5, 9, 13,...
Verified step by step guidance1
Identify the first term of the arithmetic sequence, denoted as \(a_1\). In this sequence, the first term is \(1\).
Determine the common difference \(d\) by subtracting the first term from the second term: \(d = 5 - 1\).
Write the formula for the general term (the \(n\)th term) of an arithmetic sequence, which is \(a_n = a_1 + (n - 1) \times d\).
Substitute the values of \(a_1\) and \(d\) into the formula to get the explicit formula for this sequence.
Use the formula to find the 20th term by substituting \(n = 20\) into the general term formula.
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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 each term after the first is found by adding a constant difference to the previous term. For example, in the sequence 1, 5, 9, 13,... the common difference is 4. Understanding this pattern is essential to formulating the general term.
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Arithmetic Sequences - General Formula
General Term Formula of an Arithmetic Sequence
The general term (nth term) of an arithmetic sequence can be expressed as 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 allows direct calculation of any term without listing all previous terms.
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Evaluating the nth Term
Once the general term formula is established, substituting the desired term number (like n=20) into the formula gives the specific term's value. This step avoids recursion and efficiently finds terms deep in the sequence.
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Related Practice
Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1/(1 · 2) + 1/(2 · 3) + 1/(3 · 4) + ... + 1/(n(n+1)) = n/(n + 1)
Textbook Question
Use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)5
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Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for an to find a7, the seventh term of the sequence. 0.0004, - 0.0004, 0.04, - 0.04, ...
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Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.
Textbook Question
Evaluate each factorial expression. 18!/16!
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