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Chapter 11: Problem 20

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=4, r=0.1 $$

Short Answer

Expert verified
The explicit formula is \(a_{n}=4 \cdot (0.1)^{n-1}\) and the first five terms of the sequence are: 4.0, 0.4, 0.04, 0.004, 0.0004.

Step by step solution

01

Determine the Explicit Formula

The sequence given is a geometric sequence. The explicit formula for a geometric sequence is generally \(a_{n}=a_{1} \cdot r^{(n-1)}\), where \(a_{n}\) is the nth term, \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. Here, \(a_{1}=4\) and \(r=0.1\), so the formula becomes \(a_{n}=4 \cdot (0.1)^{n-1}\).
02

Generate the First Five Terms

To find the first five terms, substitute the values of \(n\) from 1 to 5 into the formula. Thus, \(a_{1}=4 \cdot (0.1)^{0}\), \(a_{2}=4 \cdot (0.1)^{1}\), \(a_{3}=4 \cdot (0.1)^{2}\), \(a_{4}=4 \cdot (0.1)^{3}\), and \(a_{5}=4 \cdot (0.1)^{4}\), which yield the sequence 4.0, 0.4, 0.04, 0.004, 0.0004.

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Key Concepts

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

Explicit Formula
An explicit formula in a geometric sequence is like a roadmap to finding any term within the sequence. It spares us the repetitive task of always starting from the beginning.
The formula itself is \( a_{n} = a_{1} imes r^{(n-1)} \), where:
  • \( a_{n} \) stands for the nth term you want to find.
  • \( a_{1} \) is the first term in your sequence.
  • \( r \) is the common ratio, a vital clue that shows you how each term relates to the previous one.
  • \( n \) is simply the position of the term within the sequence.
So, when given \( a_{1}=4 \) and \( r=0.1 \), the explicit formula becomes \( a_{n} = 4 \times (0.1)^{(n-1)} \).
This formula lets us plug in any number for \( n \) to find corresponding terms.
Geometric Progression
A geometric progression, or geometric sequence, is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.
This sequence grows or shrinks predictably. Some important features include:
  • Exponential Nature: Because we constantly multiply by a fixed ratio, geometric sequences often change rapidly.
  • Reversed Sign: If the common ratio is negative, the signs of the terms alternate.
  • Fixed Ratio: Determines how much the sequence grows or shrinks. If it鈥檚 greater than 1, the sequence grows, whereas if it's between 0 and 1, it shrinks.
In our example, the first five terms using \( a_{n}=4 \times (0.1)^{n-1} \) are: 4.0, 0.4, 0.04, 0.004, and 0.0004, illustrating a shrinking progression.
Common Ratio
The common ratio in a geometric sequence is a cornerstone concept that directly dictates the behavior of the sequence.
Mathematically, the common ratio \( r \) is the factor that multiplies each term to get to the next. It鈥檚 calculated by dividing any term by its previous term. In our case with \( r = 0.1 \), each term is a tenth of the previous one.
  • Magnitude: If \( |r| > 1 \), terms grow larger; if \( 0 < |r| < 1 \), as in our example, terms shrink.
  • Negative Values: When \( r \) is negative, the signs of the terms alternate.
Understanding the common ratio helps predict the size and direction of a sequence's progression. It's the key that unlocks the next term from the previous, making it crucial for understanding the dynamics of geometric sequences.

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