Textbook Question
Find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a1 = 35, d = -3.
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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 21
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. 1.5, - 3, 6, -12, ...
Verified step by step guidance1
Identify the first term of the geometric sequence, denoted as \(a_1\). In this sequence, the first term is \(1.5\).
Determine the common ratio \(r\) by dividing the second term by the first term: \(r = \frac{-3}{1.5}\).
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.
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. For example, in the sequence 1.5, -3, 6, -12, ..., each term is multiplied by -2 to get the next term.
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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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Evaluating the nth Term
Once the general term formula is established, substitute the desired term number (n) into the formula to find that specific term. For example, to find the seventh term a₇, plug n = 7 into the formula and calculate the value.
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Related Practice
Textbook Question
Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...
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Textbook Question
In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. an=−2(n−1)!
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Textbook Question
In Exercises 21–22, a fair coin is tossed two times in succession. The sample space of equally likely outcomes is {HH,HT,TH,TT}. Find the probability of getting two heads.
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Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 1 · 2 + 2 · 3 + 3 · 4 + ... + n(n + 1) = n(n + 1)(n + 2)/3
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Textbook Question
Evaluate each expression.
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