Textbook Question
The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a_1=7 and a_n=a_n-1 + 5 for n≥2
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Ch. 8 - Sequences, Induction, and Probability
Chapter 9, Problem 15
In Exercises 9–16, 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 a8 when a1 = 1 000 000, r = 0.1
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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. The general form of a geometric sequence can be expressed as a_n = a_1 * r^(n-1), where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.
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4:18
Geometric Sequences - Recursive Formula
General Term Formula
The general term formula for a geometric sequence allows us to calculate any term in the sequence based on its position. Specifically, the nth term can be calculated using the formula a_n = a_1 * r^(n-1). This formula is essential for determining specific terms in the sequence, such as the 8th term in this case.
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7:17Writing a General Formula
Common Ratio
The common ratio in a geometric sequence is the factor by which we multiply each term to get the next term. It is denoted by 'r' and can be found by dividing any term by its preceding term. Understanding the common ratio is crucial for applying the general term formula correctly and predicting the behavior of the sequence.
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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
In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a6 when a₁ = 13, d = 4.
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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. a_1=3 and a_n=4a_n-1 for n≥2
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Textbook Question
In Exercises 16–18, find the indicated term of the arithmetic sequence with first term, , and common difference, d. Find a12 when a1 = -8, d = -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 a50 when a₁ = 7, d = 5.
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