Problem 20 - Algebra 2

Question

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

Step-by-Step Solution

Answer

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 1: 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 \(a_{n}=4 \cdot (0.1)^{n-1}\).

Step 2: 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.

Key Concepts

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’s 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’s 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.