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 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, ...
Verified step by step guidance1
Identify the first term of the geometric sequence, denoted as \(a_1\). In this sequence, the first term is \(0.0004\).
Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-0.0004}{0.0004}\).
Write the general formula for the \(n\)th term of a geometric sequence: \(a_n = a_1 \times r^{n-1}\).
Substitute the values of \(a_1\) and \(r\) into the formula to express \(a_n\) explicitly for this sequence.
To find the seventh term \(a_7\), substitute \(n = 7\) into the formula: \(a_7 = a_1 \times r^{6}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Geometric Sequence
A geometric sequence is a list 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 or negative, and it determines the pattern of growth or decay in the sequence.
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Geometric Sequences - Recursive Formula
General Term Formula of a Geometric Sequence
The general term (nth term) of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term, r is the common ratio, and n is the term number. This formula allows you to find any term in the sequence without listing all previous terms.
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Finding Specific Terms Using the General Formula
Once the general term formula is established, you can find any specific term by substituting the term number n into the formula. For example, to find the seventh term a₇, plug in n = 7 and calculate using the known first term and common ratio.
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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)
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Textbook Question
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,...
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Textbook Question
Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
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Textbook Question
Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.
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Textbook Question
Evaluate each factorial expression. 18!/16!
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