Problem 27 - College Algebra with Corequisite Support

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 \(a_n = -0.3 \cdot a_{n-1}\) for \(n \geq 2\).

Step 1: Identify the first term

In a geometric sequence, the first term is denoted by \(a_1\). In this sequence, the first term \(a_1\) is 10.

Step 2: Determine the common ratio

To find the common ratio \(r\) in the sequence, divide the second term by the first term. Here, the second term is \(-3\) and the first term is \(10\). So, \(r = \frac{-3}{10} = -0.3\). Verify by dividing the third term by the second term and so on to ensure consistency.

Step 3: Write the recursive formula

A recursive formula for a geometric sequence is expressed as: \(a_n = r \cdot a_{n-1}\). Here \(r = -0.3\) and the sequence starts at \(a_1 = 10\). Thus, the recursive formula is \(a_n = -0.3 \cdot a_{n-1}\) for \(n \geq 2\).

Key Concepts

Recursive Formula

In the world of mathematics, specifically sequences, a "recursive formula" plays a key role. A recursive formula allows us to determine the current term of a sequence based on the previous term, creating a step-by-step process to define a sequence. This type of formula is valuable because it simplifies the process of continuing a pattern.
Let's take a closer look at how this works in a geometric sequence. For any geometric sequence, the expression typically is of the form \(a_n = r \cdot a_{n-1}\), where:

• \(a_n\) is the nth term of the sequence,
• \(a_{n-1}\) is the (n-1)th term,
• \(r\) represents the common ratio.

For our example sequence, the recursive formula is found to be \(a_n = -0.3 \cdot a_{n-1}\). This formula indicates each term is obtained by multiplying the previous term by \(-0.3\).
By knowing the recursive formula, you gain the ability to determine any term in the sequence using the value of the previous term.

Common Ratio

The concept of a "common ratio" is central to understanding geometric sequences. To define it simply, the common ratio is the constant factor between consecutive terms in a geometric sequence. It provides the multiplying value applied to one term to get the next term.
In our example, to determine the common ratio of the sequence \(\{10, -3, 0.9, -0.27, \ldots\}\), we divided the second term \(-3\) by the first term \(10\), resulting in a ratio of \(-0.3\). This same common ratio is verified by performing the division operation between subsequent terms to ensure consistency.
Understanding the common ratio is crucial because it governs the behavior and direction (either increasing or decreasing) of the entire sequence.

First Term

In the initiation of any sequence, the "first term" is fundamental. It sets the starting point from which all other terms are derived. The first term is typically represented as \(a_1\).
In our specific sequence \(\{10, -3, 0.9, -0.27, \ldots\}\), the first term is given as \(a_1 = 10\). This number essentially seeds the sequence, and every other term is derived based on this value using the recursive formula.
Knowing \(a_1\) is crucial because it, along with the common ratio, dictates the entire course and the values of the sequence. Without it, establishing the sequence pattern would be impossible.

Sequence Consistency

Maintaining "sequence consistency" is essential to ensure that a pattern is correctly followed in a geometric sequence. Consistency means each term reliably follows the rule established by the common ratio. This reliability is checked by confirming that the ratio between consecutive terms remains constant.
For our sequence example \(\{10, -3, 0.9, -0.27, \ldots\}\), consistency is validated by confirming that each successive term results from multiplying the common ratio \(-0.3\) with its preceding term. For instance, \(-3 \times -0.3 = 0.9\).
Recognizing and ensuring consistency allows mathematicians and students alike to trust and predict the continuation of the sequence without calculating each term from scratch repeatedly. This verification forms the backbone of working with recursive formulas efficiently.