Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for each geometric sequence. \(a_{n}=\left\\{\frac{1}{512},-\frac{1}{128}, \frac{1}{32},-\frac{1}{8}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The recursive formula is \( a_{n} = a_{n-1} \times (-4) \) with \( a_1 = \frac{1}{512} \).

Step 1: Identify the First Term

The geometric sequence begin with the term \( a_1 = \frac{1}{512} \). This is the first term we need for the recursive formula.

Step 2: Find the Common Ratio

Divide the second term by the first term to find the common ratio: \( r = \frac{-\frac{1}{128}}{\frac{1}{512}} = -4 \). This indicates that each term is obtained by multiplying the previous term by \(-4\).

Step 3: Write the Recursive Formula

The recursive formula for a geometric sequence can be expressed as: \[ a_{n} = a_{n-1} \times r \] Substituting the first term and the common ratio, the formula becomes \[ a_{n} = a_{n-1} \times (-4) \] with the initial condition \( a_1 = \frac{1}{512} \).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant called the common ratio. This sequence has a specific pattern that is easy to identify once you know the common ratio. For example, if we start with 1 and our common ratio is 2, our sequence would be: 1, 2, 4, 8, 16, and so on.
Geometric sequences are useful in many areas of mathematics and science because they model situations where things grow or decay at a consistent rate. These sequences are predictable and can be calculated easily once the initial values are known.

First Term

The first term in a geometric sequence is crucial because it is the starting point for the entire sequence. In mathematical notation, the first term is often represented as \( a_1 \).
Identifying the first term is important when you're writing a formula for a sequence or trying to find any specific term in that sequence. For example, in our exercise, the first term \( a_1 \) is \( \frac{1}{512} \). This set the stage for how the rest of the sequence unfolds when combined with the common ratio.
Without knowing the first term, you cannot reliably generate the rest of the sequence or write a formula that represents it.

Common Ratio

The common ratio in a geometric sequence is a fixed value that you multiply by to get from one term to the next. It is the defining feature of a geometric sequence.
To find the common ratio, you simply divide the second term by the first term. In our example, the common ratio \( r \) is calculated by dividing \( -\frac{1}{128} \) by \( \frac{1}{512} \), resulting in \( -4 \). This means each term is obtained by multiplying the previous term by \(-4\).
The common ratio not only tells us how the sequence progresses but also indicates the direction of progression—whether it grows or shrinks, and whether it flips sign due to a negative ratio.

Geometric Progression

A geometric progression is another term for a geometric sequence, highlighting its nature of progressing from one term to the next through multiplication by a common ratio.
In a geometric progression, each term after the first is the product of the previous term and the common ratio. For instance, if the first term is \( \frac{1}{512} \) and the common ratio is \(-4\), the sequence is \( \frac{1}{512}, -\frac{1}{128}, \frac{1}{32}, -\frac{1}{8}, \ldots \).
The recursive formula for a geometric progression like this is written by using the formula \( a_n = a_{n-1} \times r \). This allows you to calculate any term in the sequence if you know the previous term. This is useful, as you can forecast any value within the sequence given the first term and the common ratio.