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Chapter 8: Problem 38

Write a recursive rule for the sequence. $$ a_n=\frac{1}{4}(5)^{n-1} $$

Short Answer

Expert verified
So the recursive rule for the sequence is \(a_1=\frac{1}{4}\) and for \(n>1\), \(a_n=\frac{5}{4}a_{n-1}\)

Step by step solution

01

Identify the initial term (a_1)

The initial term (\(a_1\)) of the sequence is obtained by plugging n=1 into the sequence formula. So, \(a_1=\frac{1}{4}(5)^{1-1}=\frac{1}{4}\)
02

Identify the common ratio

The common ratio of the sequence is the factor between consecutive terms, which in this case is 5/4 given by the formula \(\frac{1}{4}(5)^{n-1}\).
03

Formulate the recursive rule

A recursive rule defines the terms of a sequence using previous terms. In the case of a geometric sequence, any term is the product of the previous term and the common ratio. Therefore, for \(n>1\), the recursive rule is \(a_n=\frac{5}{4}a_{n-1}\)

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Key Concepts

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

Geometric Sequence
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of sequence can be readily identified by its exponential pattern, which makes it distinct.

For instance, in the sequence given by the formula \(a_n=\frac{1}{4}(5)^{n-1}\), each term is obtained by multiplying the preceding term by 5. This consistent multiplicand is an essential characteristic of a geometric sequence. Understanding this fundamental nature helps students recognize such patterns in different contexts, which is critical for solving problems effectively.
Common Ratio
The common ratio in a geometric sequence is the constant factor or ratio between any term and the previous one. It is a pivotal component that defines the behavior of the sequence.

In the sequence \(a_n=\frac{1}{4}(5)^{n-1}\), the ratio between a subsequent term and its predecessor is always 5/4 or 1.25. This tells us how each term scales relative to the previous one. To identify the common ratio, we often divide any term by the term just before it. Knowing the common ratio is essential for writing sequences in both explicit (general formula) and recursive forms.
Sequence Formula
The sequence formula provides a method to find the nth term of a sequence by a direct calculation, without the necessity of knowing the previous terms. This is known as the explicit formula.

For a geometric sequence, the explicit formula is typically expressed as \(a_n=a_1 \times r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio. In the example given, \(a_1\) is 1/4, and \(r\) is 5, serving as the base in the expression \(5^{n-1}\). By understanding and applying the sequence formula, students can swiftly determine any term in the sequence, making it an indispensable tool in sequence-related exercises.

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Most popular questions from this chapter

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

write a rule for the nth term of the arithmetic sequence. \(a_{12}=-38, a_{19}=-73\)

Describe how doubling each term in an arithmetic sequence changes the common difference of the sequence. Justify your answer.

USING EQUATIONS One term of an arithmetic sequence is \(a_8=-13\). The common difference is \(-8\). What is a rule for the \(n\)th term of the sequence? (A) \(a_n=51+8 n\) (B) \(a_n=35+8 n\) (C) \(a_n=51-8 n\) (D) \(a_n=35-8 n\)

CRITICAL THINKING One of the major sources of our knowledge of Egyptian mathematics is the Ahmes papyrus, which is a scroll copied in 1650 B.C. by an Egyptian scribe. The following problem is from the Ahmes papyrus. Divide 10 hekats of barley among 10 men so that the common difference is \(\frac{1}{8}\) of a hekat of barley. Use what you know about arithmetic sequences and series to determine what portion of a hekat each man should receive.
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