Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n2
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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 13
Write the first six terms of each arithmetic sequence. an = an-1 -0.4, a1 = 1.6
Verified step by step guidance1
Identify the first term of the arithmetic sequence, which is given as \(a_1 = 1.6\).
Recognize that the common difference \(d\) is the amount subtracted each time, so \(d = -0.4\).
Use the recursive formula \(a_n = a_{n-1} + d\) to find each subsequent term by subtracting 0.4 from the previous term.
Calculate the second term: \(a_2 = a_1 - 0.4\).
Continue this process to find the third, fourth, fifth, and sixth terms by repeatedly subtracting 0.4 from the previous term.
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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. This difference can be positive, negative, or zero, and it defines the pattern of the sequence.
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Arithmetic Sequences - General Formula
Recursive Formula for Sequences
A recursive formula defines each term of a sequence using the previous term(s). In this problem, the formula an = an-1 - 0.4 means each term is 0.4 less than the term before it, starting from the initial term a₁.
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Finding Terms of a Sequence
To find terms of a sequence using a recursive formula, start with the given first term and repeatedly apply the formula to find subsequent terms. For example, subtract 0.4 from each term to get the next, until the desired number of terms is found.
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Related Practice
Textbook Question
In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2
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Textbook Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1 and common ratio, r. Find a40 when a1 = 1000, r = - 1/2
Textbook Question
Use the formula for nCr to evaluate each expression. 5C0
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Textbook Question
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a1=7 and an=an-1 + 5 for n≥2
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