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Chapter 11: Problem 139

Find the \(5^{\text { th }}\) term of the arithmetic sequence \(\\{9 b, 5 b, b, \ldots\\}\)

Short Answer

Expert verified
The 5th term is -7b.

Step by step solution

01

Identify the First Term

In the arithmetic sequence \(\{9b, 5b, b, \ldots\}\), the first term \(a_1\) is \(9b\).
02

Determine the Common Difference

To find the common difference \(d\), subtract the first term from the second term: \(5b - 9b = -4b\). Thus, the common difference \(d = -4b\).
03

Use the Formula for the Nth Term

The formula to find the \(n^{\text{th}}\) term of an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\).
04

Plug Values into the Formula

Substitute \(a_1 = 9b\), \(d = -4b\), and \(n = 5\) into the formula: \(a_5 = 9b + (5-1) \cdot (-4b)\).
05

Simplify the Expression

Simplify the expression: \(a_5 = 9b + 4 \cdot (-4b) = 9b - 16b = -7b\).

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Key Concepts

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

First Term
In an arithmetic sequence, the first term is simply the starting point of the sequence. It is often denoted by \(a_1\). In the sequence \(\{9b, 5b, b, \ldots\}\), the first term \(a_1\) is \(9b\). This means that the sequence begins with \(9b\) and each subsequent term is derived based on this starting value. The importance of identifying the first term lies in its role as the foundation of the sequence. By knowing \(a_1\), you can start calculating other terms once you have the common difference. This step is essential for progressing through any sequence and accurately determining later terms.
Common Difference
The common difference is a key feature of arithmetic sequences. It is the difference between consecutive terms and is denoted by \(d\). For the sequence \(\{9b, 5b, b, \ldots\}\), calculate \(d\) by subtracting the first term from the second term: \(5b - 9b = -4b\).
The common difference indicates how much we add to the previous term to get the next term. In our example, \(d = -4b\), meaning we subtract \(4b\) in each step moving forward in the sequence. Understanding the common difference helps you grasp how the sequence is constructed and predict future terms. It's essentially the magic number that keeps the sequence going.
Nth Term Formula
The nth term formula of an arithmetic sequence is the rule that allows you to find any term in the sequence without needing to list all preceding terms. The formula is given by \(a_n = a_1 + (n-1) \cdot d\), where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.
In our exercise, to find the \(5^{\text{th}}\) term, \(a_5\), substitute \(a_1 = 9b\), \(d = -4b\), and \(n = 5\) into the formula:
\[a_5 = 9b + (5-1) \cdot (-4b)\]
Simplify to find:
\[a_5 = 9b - 16b = -7b\]
This formula is invaluable in determining any term in a sequence quickly, and it encapsulates the entire structure of the sequence within one equation.

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