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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

The rule for a recursive sequence is as follows. $$ \begin{aligned} &f(1)=3, f(2)=10 \\ &f(n)=4+2 f(n-1)-f(n-2) \end{aligned} $$ a. Write the first five terms of the sequence. b. Use finite differences to find a pattern. What type of relationship do the terms of the sequence show? c. Write an explicit rule for the sequence.

The Sierpinski carpet is a fractal created using squares. The process involves removing smaller squares from larger squares. First, divide a large square into nine congruent squares. Then remove the center square. Repeat these steps for each smaller square, as shown below. Assume that each side of the initial square is 1 unit long. a. Let \(a_n\) be the total number of squares removed at the \(n\)th stage. Write a rule for \(a_n\). Then find the total number of squares removed through Stage 8 . b. Let \(b_n\) be the remaining area of the original square after the \(n\)th stage. Write a rule for \(b_n\). Then find the remaining area of the original square after Stage 12 .

The variables x and y vary inversely. Use the given values to write an equation relating x and y.Then fi nd y when x = 4. $$ x=2, y=9 $$

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) \(a_n=35-8 n\)

In Exercises 5鈥�12, tell whether the sequence is geometric. Explain your reasoning. \(729,243,81,27,9, \ldots\)
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