Textbook Question
A statement Sn about the positive integers is given. Write statements Sk and Sk+1 simplifying statement Sk+1 completely. Sn: 4 + 8 + 12 + ... + 4n = 2n(n + 1)
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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 5
Write the first five terms of each geometric sequence. an = - 4a(n-1), a1 = 10
Verified step by step guidance1
Identify the first term of the geometric sequence, which is given as \(a_1 = 10\).
Recognize the recursive formula for the sequence: \(a_n = -4a_{n-1}\), meaning each term is obtained by multiplying the previous term by \(-4\).
Calculate the second term using the formula: \(a_2 = -4 \times a_1\).
Calculate the third term using the formula: \(a_3 = -4 \times a_2\).
Continue this process to find the fourth and fifth terms: \(a_4 = -4 \times a_3\) and \(a_5 = -4 \times a_4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Sequence Definition
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This ratio can be positive, negative, or fractional, and it determines the pattern of the sequence.
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4:18
Geometric Sequences - Recursive Formula
Recursive Formula for Sequences
A recursive formula defines each term of a sequence using one or more previous terms. In this problem, the term a_n depends on the previous term a_(n-1) multiplied by -4, which means each term is generated by applying this rule starting from the initial term.
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Finding Terms of a Sequence
To find terms of a sequence given a recursive formula and the first term, start with the initial term and repeatedly apply the formula to find subsequent terms. For example, multiply the first term by the common ratio to get the second term, then use the second term to find the third, and so on.
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Related Practice
Textbook Question
Write the first six terms of each arithmetic sequence.
Textbook Question
A statement Sn about the positive integers is given. Write statements S1, S2 and S3 and show that each of these statements is true. Sn: 3 is a factor of n3 - n.
Textbook Question
Write the first six terms of each arithmetic sequence. a1 = 300, d= -90
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Textbook Question
Write the first four terms of each sequence whose general term is given. an=(−3)n
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Textbook Question
Use the formula for nPr to evaluate each expression. 6P6