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Chapter 9: Problem 53

Write the first five terms of the sequence defined recursively. $$a_{1}=28, a_{k}=a_{k-1}-4$$

Short Answer

Expert verified
The first five terms of the sequence are: 28, 24, 20, 16, 12.

Step by step solution

01

Understanding the first term

The first term of the sequence is given as \(a_1=28\).
02

Calculate the second term

Using the recursive formula \(a_{k}=a_{k-1}-4\), substituting k with 2 gives us \(a_2= a_{2-1}-4\), which simplifies to: \(a_2= a_{1}-4\). As \(a_1=28\), so \(a_2=28-4=24\).
03

Calculate the third term

Following the recursive formula again, substituting k with 3, \(a_{3}=a_{3-1}-4\). Simplifying it gives: \(a_{3}=a_{2}-4\). Using the value of \(a_2\) calculated in step 2 (24), \(a_3=24-4=20\).
04

Calculate the fourth term

Following the same procedure, we substitute k with 4 to get \(a_{4}=a_{4-1}-4\). Simplifying it gives: \(a_{4}=a_{3}-4\). Using the value of \(a_3\) calculated in step 3 (20), \(a_4=20-4=16\).
05

Calculate the fifth term

Lastly, substituting k with 5 in the recursive equation, we get \(a_{5}=a_{5-1}-4\). This simplifies to: \(a_{5}=a_{4}-4\). Using the value of \(a_4\) calculated in step 4 (16), \(a_5=16-4=12\).

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Key Concepts

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

Sequence Definition
A sequence in mathematics is a set of numbers arranged in a specific order based on a certain rule. Each number in this order is known as a term. Sequences come in various forms, with rules that can be simple or complex. The rule is essential as it defines the relationship between the terms.

A sequence can be finite, with a limited number of terms, or infinite, extending indefinitely. Mathematicians use sequences to study patterns and apply them in disciplines ranging from statistics to computer science. Understanding the initial terms and the rule is critical, as they help to predict future terms or scrutinize the nature of the sequence.
Arithmetic Sequence
An arithmetic sequence is one of the most straightforward types of sequence, where each term is obtained by adding a consistent number, called the common difference, to the previous term. This difference can be positive or negative and is constant throughout the arithmetic sequence.

For example, if the difference is 3, starting with the first term as 2, the sequence would be 2, 5, 8, 11, and so forth. One of the advantages of an arithmetic sequence is its predictability; once you know the first term and the common difference, you can easily calculate any term in the sequence without necessarily knowing all the prior terms.
Recursive Formula
A recursive formula is a way of defining the terms of a sequence with respect to the preceding terms. Unlike an explicit formula, which directly calculates any term in a sequence, a recursive formula builds each term from the previous one(s).

To effectively use a recursive formula, you typically need to know one or more initial terms. Recursion is a fundamental concept not only in mathematics but also in computer science, where functions call themselves to solve complex problems step by step.

In the given exercise, the recursive formula is used to find the terms of an arithmetic sequence by subtracting 4 from the previous term. This is a simple yet powerful way to generate a sequence methodically by feeding each term's value back into the formula to find the next one.

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Most popular questions from this chapter

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=6, r=-\frac{1}{4}$$

Write the first five terms of the sequence defined recursively. $$a_{0}=1, a_{1}=3, a_{k}=a_{k-2}+a_{k-1}$$

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$\frac{1}{5}, \frac{2}{7}, \frac{3}{9}, \frac{4}{11}, \dots$$

Simplify the factorial expression. $$\frac{5 !}{7 !}$$

Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio.Identifying a Geometric Sequence Determine whether or not the sequence is geometric. If it is, find the common ratio. $$1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \dots$$
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