Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, write the first five terms of the geometric sequence. \(a_{1}=7, \quad a_{n}=0.2 a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms are 7, 1.4, 0.28, 0.056, 0.0112.

Step 1: Understanding the Problem Statement

We are given a geometric sequence where the first term is \(a_1 = 7\) and the relationship between consecutive terms is given by \(a_n = 0.2 \cdot a_{n-1}\). Our task is to find the first five terms of this sequence.

Step 2: Calculate the First Term

The first term \(a_1\) is already provided as \(7\).

Step 3: Calculate the Second Term

Using the formula \(a_2 = 0.2 \cdot a_1\), substitute \(a_1 = 7\):\[a_2 = 0.2 \cdot 7 = 1.4\]

Step 4: Calculate the Third Term

Use the formula \(a_3 = 0.2 \cdot a_2\), where \(a_2 = 1.4\):\[a_3 = 0.2 \cdot 1.4 = 0.28\]

Step 5: Calculate the Fourth Term

Apply the same formula, \(a_4 = 0.2 \cdot a_3\), using \(a_3 = 0.28\):\[a_4 = 0.2 \cdot 0.28 = 0.056\]

Step 6: Calculate the Fifth Term

Finally, apply the formula \(a_5 = 0.2 \cdot a_4\), where \(a_4 = 0.056\):\[a_5 = 0.2 \cdot 0.056 = 0.0112\]

Key Concepts

Sequence Terms

In mathematics, a sequence is a set of numbers arranged in a specific order. These numbers are called "terms". In a geometric sequence, each term is derived from the previous one by multiplying it with a constant number.
This number is known as the "common ratio".

- The first term in a sequence is typically denoted as \(a_1\), and in our example, it is given as \(7\).- Each subsequent term can be derived by multiplying the previous term with the common ratio \(0.2\).

To find the terms of a sequence:

• Start with the initial term, which is given.
• Multiply the last known term by the common ratio to find the next term.

This method continues until you find as many terms as needed. It’s essential to calculate each term correctly to ensure the entire sequence follows the pattern established by the common ratio.

Recurrence Relation

A recurrence relation is an equation defining each term of a sequence as a function of its preceding term(s). In other words, it provides a rule for determining the current term given one or more previous terms.
In our geometric sequence, the recurrence relation is given as \(a_n = 0.2 \cdot a_{n-1}\).

To break it down:

• \(a_n\) refers to the term we want to calculate.
• \(a_{n-1}\) is the previous term in the sequence.
• \(0.2\) is the multiplicative factor, or the common ratio in this case.

This relation provides a systematic way to generate terms one after the other by consistently applying the multiplicative factor to each successive term that you identify.

Multiplicative Factor

The multiplicative factor is a critical component in geometric sequences. It is the constant number by which each term is multiplied to get the next term. Here, the multiplicative factor is \(0.2\), often referred to as the "common ratio" in the context of geometric sequences.

The Impact of the Multiplicative Factor

The value of the multiplicative factor affects how rapidly the terms in the sequence change:

• If the factor is greater than \(1\), the terms increase.
• If it is between \(0\) and \(1\), like our example of \(0.2\), the terms decrease.
• A negative factor would cause the sequence to alternate between positive and negative terms.

Choosing the right multiplicative factor is crucial as it establishes the foundational growth or decay pattern characterizing the geometric sequence.