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Chapter 8: Problem 11

Write a recursive rule for the sequence. $$ 21,14,7,0,-7, \ldots $$

Short Answer

Expert verified
The recursive rule for the given sequence is \(a_1 = 21\) and \(a_{n} = a_{n-1} - 7\) for \(n \geq 2\).

Step by step solution

01

Identify the pattern

Look at the given sequence: \(21,14,7,0,-7, \ldots\). Observe carefully that each term is 7 less than the previous one. This shows that the sequence follows a decreasing pattern.
02

Formulate the recursive rule

With the identified pattern, formulate the recursive rule. The first term (termed \(a_1\)) of the sequence is 21. Each subsequent term (termed as \(a_{n}\)) is 7 less than the preceding term (\(a_{n-1}\)). Therefore, the recursive rule can be defined as: \(a_1 = 21\) and \(a_{n} = a_{n-1} - 7\) for \(n \geq 2\). This means, the first term is 21 and each term after the first is 7 less than the previous term.

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Key Concepts

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

Arithmetic Sequences
Arithmetic sequences are a fundamental concept in mathematics, particularly in algebra. They consist of a series of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This common difference can be positive, negative, or even zero, which would make all the terms of the sequence equal.

For instance, in the sequence \( 21, 14, 7, 0, -7, \ldots \), the common difference is -7 because each term is 7 less than the term before it. Arithmetic sequences are everywhere in real life, from calculating bank interest to predicting the number of seats a party might win in an election based on trends.
Recursive Formulas
Recursive formulas offer a way to express sequences where each term is defined in relation to one or more previous terms. This stands in contrast to explicit formulas, where each term is defined independently from the others.

In our example \(a_1 = 21\) and \(a_{n} = a_{n-1} - 7\) for \(n \textgreater= 2\), the recursive rule clearly shows that knowledge of the preceding term provides a pathway to finding any subsequent terms in the sequence. These formulas are particularly useful in computer science, for programming loops, and in finance for modeling compound interest or investments.
Sequence Patterns
Recognizing sequence patterns is a skill that allows one to identify the rule that defines the progression of elements in a sequence. It's about finding the logic behind the arrangement of numbers, which can help in predicting future terms or even in understanding deeper mathematical structures.

By observing that the pattern of the sequence \(21, 14, 7, 0, -7, \ldots \) involves subtracting the same number repeatedly, it's easy to determine that we are dealing with an arithmetic sequence marked by a constant negative difference. This pattern recognition is not only key for mathematics but it is also used in areas such as data analysis, cryptography, and even music theory.

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Most popular questions from this chapter

In Exercises 23-30, write a rule for the \(n\)th term. Then graph the first six terms of the sequence. $$ a_3=27, r=3 $$

The \(n\)th term of a geometric sequence has the form \(a_n=\) ___________.

write a rule for the nth term of the sequence. Then graph the fi rst six terms of the sequence. \(a_{21}=-25, d=-\frac{3}{2}\)

In Exercises 23-30, write a rule for the \(n\)th term. Then graph the first six terms of the sequence. $$ a_5=3, r=-\frac{1}{3} $$

In Exercises 47鈥�52, find the sum. $$ \sum_{i=1}^8 5\left(\frac{1}{3}\right)^{i-1} $$
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