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Chapter 10: Problem 7

Write the first five terms of each geometric sequence. $$a_{n}=-5 a_{n-1}, \quad a_{1}=-6$$

Short Answer

Expert verified
The first five terms of the sequence are -6, 30, -150, 750, -3750

Step by step solution

01

Identify the first term

Given that \(a_{1}\) is -6, this is the first term of the sequence. So, Term 1 = -6.
02

Calculate the second term

Substitute \(n = 2\) into the recurrence definition, \(a_{n}=-5 a_{n-1}\), to find the second term. So, \(a_{2} = -5 \cdot a_{1}\), which gives -5*(-6) = 30. Therefore, Term 2 = 30.
03

Calculate the third term

Substitute \(n = 3\) into the recurrence definition, \(a_{n}=-5 a_{n-1}\), to find the third term. So, \(a_{3} = -5 \cdot a_{2}\), which gives -5*30 = -150. Therefore, Term 3 = -150.
04

Calculate the fourth term

Substitute \(n = 4\) into the recurrence definition, \(a_{n}=-5 a_{n-1}\), to find the fourth term. So, \(a_{4} = -5 \cdot a_{3}\), which gives -5*(-150) = 750. Therefore, Term 4 = 750.
05

Calculate the fifth term

Substitute \(n = 5\) into the recurrence definition, \(a_{n}=-5 a_{n-1}\), to find the fifth term. So, \(a_{5} = -5 \cdot a_{4}\), which gives -5*750 = -3750. Therefore, Term 5 = -3750.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Sequence Definition
A geometric sequence is a series 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. This ratio remains constant throughout the sequence, which creates a pattern of multiplication that gives the sequence its 'geometric' nature.

For example, if we have an initial term, known as the first term of the sequence, and we are given a common ratio, we can generate the subsequent terms of the sequence using this ratio. It is this repetitive process of multiplying by a common ratio that characterizes a geometric sequence.
Recurrence Relation
The recurrence relation is a way of defining the terms of a sequence with respect to the previous terms. In essence, it provides a recursive formula that can be used to calculate any term in the sequence, given that the initial term or terms are known. This relation captures the essence of a geometric sequence through the formula:
\[ a_{n} = r \times a_{n-1} \]
where \( a_{n} \) is the nth term, \( a_{n-1} \) is the term before it, and \( r \) is the common ratio. The recurrence relation adds an element of simplicity to constructing the sequence because it articulates an easy-to-follow rule which, when applied iteratively, determines the entire sequence.
Sequence Term Calculation
The term calculation of a geometric sequence is a methodological process. Given the first term and the common ratio, we can calculate any term in the sequence. The nth term of a geometric sequence can be found using the formula:
\[ a_{n} = a_{1} \times r^{(n-1)} \]
where \( a_{1} \) is the first term, \( r \) is the common ratio, and \( n \) represents the term's position in the sequence.

In the given exercise, the initial term and recurrence relation would allow students to calculate any term of the sequence. This approach not only leads to understanding each term but also illustrates the underpinnings of geometric progression.

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