Problem 30 - Precalculus with Limits

Question

In Exercises 29 - 34, write the first five terms of the geometric sequence. Determine the common ratio and write the \( n \)th term of the sequence as a function of \( n \). \( a_1 = 81, a_{k + 1} = \dfrac{1}{3} a_k \)

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are {81, 27, 9, 3, 1}. The nth term of the sequence is given by \( a_n = 81 * \dfrac{1}{3}^{n-1} \).

Step 1: Identify the first term and common ratio

From the provided exercise, we are informed that the first term \( a_1 = 81 \) and the common ratio is such that \( a_{k+1} = \dfrac{1}{3}a_k \), which implies that the common ratio \( r = \dfrac{1}{3} \). This ratio is the factor by which each term multiplies to get to the next term.

Step 2: Generate the first five terms of the sequence

To generate the first five terms of the sequence we will multiply each term by the common ratio. As given, the first term \( a_1 = 81 \). The second term will therefore be \( a_2 = a_1 * r = 81 * \dfrac{1}{3} = 27 \), the third term is \( a_3 = a_2 * r = 27 * \dfrac{1}{3} = 9 \), the fourth term is \( a_4 = a_3 * r = 9 * \dfrac{1}{3} = 3 \), and the fifth term is \( a_5 = a_4 * r = 3 * \dfrac{1}{3} = 1 \). Thus, the first five terms of the sequence are {81, 27, 9, 3, 1}.

Step 3: Write the nth term of the sequence

The general form of the nth term of a geometric sequence, \( a_n \) is given by \( a_n = a_1 * r^{n-1} \). Therefore, for our sequence, \( a_n = 81 * \dfrac{1}{3}^{n-1} \). This gives the nth term as a function of \( n \) in this geometric sequence.

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a fundamental aspect. It represents the factor by which we multiply a term to get the next term in the sequence. For the sequence determined in the exercise, the common ratio is \( \frac{1}{3} \). This means that each subsequent term is one third of the previous term. When identifying the common ratio, note that if this value is less than one, the sequence terms will decrease, as in this exercise. Conversely, a common ratio greater than one results in an increasing sequence. Understanding the common ratio helps us predict the behavior of the sequence as it progresses.

Nth Term

The nth term is a formula that allows us to find any term in a geometric sequence without listing all the preceding terms. For any geometric sequence, the nth term \( a_n \) can be represented by the formula \( a_n = a_1 \cdot r^{n-1} \). In our exercise, where \( a_1 = 81 \) and the common ratio \( r = \frac{1}{3} \), the formula becomes:

• \( a_n = 81 \cdot \left(\frac{1}{3}\right)^{n-1} \)

This formula allows us to calculate the value of the nth term directly. For example, substituting \( n = 3 \) gives \( a_3 = 81 \cdot \left(\frac{1}{3}\right)^2 = 9 \). This application of the nth term formula makes it a powerful tool in sequences.

Sequence Terms

Sequence terms are the individual numbers or elements that make up a sequence. In a geometric progression, each term relates to the previous term through multiplication by the common ratio. Given the problem, the first term \( a_1 = 81 \), and using the common ratio \( \frac{1}{3} \), we generated the first five terms as follows:

• \( a_1 = 81 \)
• \( a_2 = 27 \)
• \( a_3 = 9 \)
• \( a_4 = 3 \)
• \( a_5 = 1 \)

Each term here is exactly one third of its predecessor, illustrating the consistent nature of geometric sequences. Recognizing the pattern in sequence terms aids in understanding the sequence as a whole.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This sequence can grow or shrink, depending on whether the ratio is greater than or less than one. In this exercise, we have a geometric progression that decreases as the common ratio is \( \frac{1}{3} \). The terms form a clear pattern of multiplication by a constant factor across the sequence. Recognizing a geometric progression allows for easy prediction of future terms using the nth term formula. This concept is central to understanding how geometric sequences function and maintaining consistency across calculated terms.