Problem 27 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\\{10,-3,0.9,-0.27, \ldots\\} $$

Step-by-Step Solution

Answer

The recursive formula is \( a_n = a_{n-1} \, \cdot \, (-0.3) \) for \( n > 1 \), with \( a_1 = 10 \).

Step 1: Identify the first term

The first term of the sequence, \( a_1 \), is given as \( 10 \). This term is crucial for defining the initial part of the recursive formula.

Step 2: Determine the common ratio

To find the common ratio \( r \) of a geometric sequence, divide the second term by the first term: \( r = \frac{-3}{10} = -0.3 \). Verify this by checking that \( r = \frac{0.9}{-3} = -0.3 \) and \( r = \frac{-0.27}{0.9} = -0.3 \). Thus, the common ratio \( r \) is consistent.

Step 3: Write the recursive formula

The recursive formula for a geometric sequence is \( a_n = a_{n-1} \, \cdot \, r \). Since \( a_1 = 10 \) and \( r = -0.3 \), the recursive formula becomes \( a_n = a_{n-1} \, \cdot \, (-0.3) \) for \( n > 1 \).

Key Concepts

Geometric Sequence

A geometric sequence is a sequence of numbers where each term (except the first) is a product of the previous term and a constant called the common ratio. It has a distinctive pattern that multiples the preceding term by this fixed number.
In a geometric sequence, if the first term is known and the common ratio is given, you can easily calculate all subsequent terms.
More formally, if the first term is labeled as \( a_1 \) and the common ratio is \( r \), then the sequence can be described as:

• \( a_1 \)
• \( a_2 = a_1 \times r \)
• \( a_3 = a_2 \times r \), and so on.

Geometric sequences are widely used in various fields, including finance, computer science, and physics, due to their cumulative growth patterns.

Common Ratio

The common ratio in a geometric sequence is a critical factor that determines how the sequence grows or shrinks. It is the constant factor by which you multiply each term to get the next one.
To find the common ratio \( r \), take any term and divide it by the previous term (except for the first term, since there is none before it).
In the example sequence \( \{10, -3, 0.9, -0.27, \ldots\} \), you find the common ratio by:

• \( r = \frac{-3}{10} = -0.3 \)
• Verify by checking: \( \frac{0.9}{-3} = -0.3 \) and \( \frac{-0.27}{0.9} = -0.3 \)

This consistency confirms the common ratio is \( -0.3 \). Understanding the common ratio helps predict the behavior and direction of the sequence.

First Term

The first term of a sequence is the starting point from which the sequence builds. It is the initial term and is vital to defining the rest of the sequence.
In our context, the first term \( a_1 \) is explicitly given as 10 in the sequence. Without this foundational starting point, creating the recursive formula would be impossible.
Setting the first term solidifies the sequence's structure. It is the baseline value to which the common ratio is applied. Notably, changes to the first term result in a shifted sequence, albeit maintaining the same common ratio.

Recursion in Sequences

Recursion is a powerful concept in sequences where each term is derived by looking back to one or more of the previous terms. This means that instead of looking for some external equation to determine terms, you define subsequent terms based on prior ones.
In a geometric sequence, recursion is used to write formulas that reflect this pattern. For example, given a first term and a common ratio, the recursive formula expresses how each term relates to its predecessor:

• The recursive formula: \( a_n = a_{n-1} \cdot r \)
• For our sequence, \( a_n = 10 \times (-0.3)^{(n-1)} \)

This allows you to generate every term of the sequence by applying the process repetitively, making recursion highly efficient and intuitive for understanding sequential relationships.