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Chapter 8: Problem 48

Write an explicit rule for the sequence. $$ a_1=-5, a_n=\frac{1}{4} a_{n-1} $$

Short Answer

Expert verified
The explicit rule for the sequence is \(a_n = (-5) \cdot \left(\frac{1}{4}\right)^{n - 1}\)

Step by step solution

01

Identify the components of the geometric sequence

From the given rule, it can be determined that the sequence is geometric and its first term \(a_1 = -5\) and the common ratio \(r = \frac{1}{4}\).
02

Derive the formula for an explicit rule of a geometric sequence

A geometric sequence can be expressed using the formula \(a_n = a_1 \cdot r^{n - 1}\) where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Substitute the first term and the common ratio into the formula.
03

Write the explicit rule

Using the formula from Step 2 and the given information, the explicit rule for the sequence is \(a_n = (-5) \cdot \left(\frac{1}{4}\right)^{n - 1}\)

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Key Concepts

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

Understanding the Explicit Formula
An explicit formula in the context of sequences is a way to find any term in a sequence without needing to know the previous term. This is very useful as it allows us to jump directly to the nth term without calculating all the preceding terms.

In a geometric sequence, the explicit formula is given by:
  • \( a_n = a_1 \cdot r^{n-1} \)
Here, \( a_n \) represents the nth term you want to calculate. The variable \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the position of the term within the sequence.

This formula allows you to easily calculate the value of the sequence at any point, making it a powerful tool for understanding and working with sequences.
Exploring the Common Ratio
The common ratio is a key concept in geometric sequences. It's the factor by which we multiply each term to get the next term in the sequence.

In our original exercise, the common ratio \(r\) is \(\frac{1}{4}\). This means that each term is one fourth of the previous term, or equivalently, we multiply each term by \(\frac{1}{4}\) to get the next term.

Understanding the common ratio is crucial because it dictates the growth or decay of the sequence. If the common ratio is greater than one, the sequence grows. If it is between zero and one, the sequence diminishes. This is an essential concept in identifying the nature and behavior of geometric sequences.
The Geometric Sequence Formula
The geometric sequence formula is a concise way of expressing the terms of a geometric sequence using an algebraic expression. It helps uncover the relationships between the terms, making predictions about future terms straightforward.

The general format of a geometric sequence formula is:
  • \( a_n = a_1 \cdot r^{n-1} \)
This is applicable for any geometric sequence and enables calculations of any term \( a_n \) using the starting term \( a_1 \) and the common ratio \( r \).

Once plugged into the formula, we can derive the specific rule that allows us to compute any term in the series directly, which is beneficial for efficiently solving problems without stepwise computations for each term.

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Most popular questions from this chapter

Describe the pattern, write the next term, and write a rule for the \(\boldsymbol{n}\) th term of the sequence. \(1.2,4.2,9.2,16.2, \ldots\)

Describe and correct the error in finding the sum of the series. \(\begin{aligned} \sum_{i=2}^4 i^2 &=\frac{4(4+1)(2 \cdot 4+1)}{6} \\\ &=\frac{180}{6} \\ &=30 \end{aligned}\)

A regional soccer tournament has 64 participating teams. In the first round of the tournament, 32 games are played. In each successive round, the number of games decreases by a factor of \(\frac{1}{2}\). a. Write a rule for the number of games played in the \(n\)th round. For what values of \(n\) does the rule make sense? Explain. b. Find the total number of games played in the regional soccer tournament.

fi nd the sum. \(\sum_{i=1}^{41}(-2.3+0.1 i)\)

The Sierpinski triangle is a fractal created using equilateral triangles. The process involves removing smaller triangles from larger triangles by joining the midpoints of the sides of the larger triangles as shown. Assume that the initial triangle has an area of 1 square foot. a. Let \(a_n\) be the total area of all the triangles that are removed at Stage \(n\). Write a rule for \(a_n\). b. Find \(\sum_{n=1}^{\infty} a_n\). Interpret your answer in the context of this situation.
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