Problem 31 - College Algebra

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 \( a_1 = \frac{1}{512} \); \( a_n = a_{n-1} \cdot (-4) \).

Step 1: Understand the Geometric Sequence

The sequence given is \( a_n = \left\{ \frac{1}{512}, -\frac{1}{128}, \frac{1}{32}, -\frac{1}{8}, \ldots \right\} \). This appears to be a geometric sequence because each term can be obtained by multiplying the previous term by a fixed number, known as the common ratio.

Step 2: Identify the Common Ratio

Find the ratio \( r \) between consecutive terms by dividing the second term by the first term. That is, \( r = \frac{-\frac{1}{128}}{\frac{1}{512}} = -4 \). We can double-check this for other pairs: \( \frac{\frac{1}{32}}{-\frac{1}{128}} = -4 \) as well, confirming that the common ratio is \(-4\).

Step 3: Write the Recursive Formula

The recursive formula for any geometric sequence can be written as \( a_n = a_{n-1} \cdot r \), where \( r \) is the common ratio. In this case, \( a_n = a_{n-1} \cdot (-4) \).

Step 4: Determine the Initial Condition

For a recursive formula, we also need an initial term, \( a_1 \). Given \( a_1 = \frac{1}{512} \), we can write the complete recursive formula.

Key Concepts

Recursive Formula

A recursive formula allows us to express each term of a sequence in terms of the previous term. This is particularly useful in a geometric sequence, where each term is generated by multiplying the prior term by a constant, known as the "common ratio". In our exercise, we aim to write a recursive formula for the given geometric sequence: \( a_n = \left\{ \frac{1}{512}, -\frac{1}{128}, \ldots \right\} \).
To do so, we need to find how each term relates to the previous one. This relationship is represented as:

• \( a_n = a_{n-1} \cdot r \)

In this formula, \( a_n \) represents the current term, and \( a_{n-1} \) represents the term before it.
The symbol \( r \) is the common ratio, which dictates how the sequence progresses. For our example sequence, once we know \( r \), we plug it into the formula to complete the recursive description.

Common Ratio

The common ratio is the key component that determines how a sequence progresses from one term to the next in geometric sequences. It is a fixed number that, when multiplied by a term, gives the subsequent term. To find this ratio in a sequence, you divide any term by its preceding term.
Let's apply this to our example sequence: \( a_n = \left\{ \frac{1}{512}, -\frac{1}{128}, \ldots \right\} \).
The calculation involves selecting two consecutive terms, such as \( -\frac{1}{128} \) and \( \frac{1}{512} \). When you divide the second term by the first:

• \( r = \frac{-\frac{1}{128}}{\frac{1}{512}} = -4 \)

The negative sign indicates the sequence alternates in sign. Consistently applying this division to other term pairs, like \( \frac{1}{32} \) and \( -\frac{1}{128} \), reaffirms the common ratio as \( -4 \). This consistency is one hallmark of a reliable geometric sequence.

Initial Term

Identifying the initial term of a sequence is crucial when constructing a recursive formula, as it serves as the starting point from which subsequent terms are generated. For our specific sequence \( a_n = \left\{ \frac{1}{512}, -\frac{1}{128}, \ldots \right\} \), the initial term is \( a_1 = \frac{1}{512} \).
This value is essential when expressing the recursive formula, as it provides the base term that begins the entire sequence using the recursive relationship.

• Without \( a_1 \), the sequence would remain incomplete because you wouldn't know where to start applying your common ratio.

Having this starting point allows us to accurately use our recursive formula of \( a_n = a_{n-1} \cdot (-4) \) to generate all subsequent terms effectively.