Textbook Question
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=4 and a_n=2a_n-1 + 3 for n≥2
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Ch. 8 - Sequences, Induction, and Probability
Chapter 9, Problem 19
In Exercises 17–24, write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for a_n to find a7, the seventh term of the sequence. 18, 6, 2, 2/3, ...
Verified step by step guidance1
Identify the first term of the sequence, denoted as a_1. In this sequence, a_1 = 18.
Determine the common ratio (r) of the geometric sequence by dividing the second term by the first term. Here, r = 6/18.
Write the formula for the nth term of a geometric sequence: a_n = a_1 * r^(n-1).
Substitute the known values into the formula: a_n = 18 * (1/3)^(n-1).
To find the seventh term (a_7), substitute n = 7 into the formula: a_7 = 18 * (1/3)^(7-1).
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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 sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 18, 6, 2, 2/3, the common ratio can be found by dividing any term by its preceding term, which in this case is 1/3.
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Geometric Sequences - Recursive Formula
General Term Formula
The general term (nth term) of a geometric sequence can be expressed using the formula a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number. This formula allows us to calculate any term in the sequence without having to list all previous terms.
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7:17Writing a General Formula
Finding Specific Terms
To find a specific term in a geometric sequence, such as a7, we substitute n with 7 in the general term formula. By calculating a7 using the identified first term and common ratio, we can determine the value of the seventh term in the sequence, which is essential for understanding the behavior of the sequence.
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Related Practice
Textbook Question
In Exercises 19–21, 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. -7, -3, 1, 5 ...
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Textbook Question
In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a200 when a₁ = −40, d = 5.
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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. a_n = n^2/n!
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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. a_n=(n+1)!/n^2
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Textbook Question
In Exercises 15–22, 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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