Problem 16 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ 144,36,9, \frac{9}{4}, \dots $$

Step-by-Step Solution

Answer

The recursive formula for the given sequence is \(a_n = \frac{a_{n-1}}{4}\) and the next term is \(\frac{9}{16}\).

Step 1: Identify the Pattern

On careful observation, it can be seen that each term is one-fourth (or divided by 4) of the preceding term. This indicates a geometric sequence where the common ratio is 1/4.

Step 2: Write the Recursive Formula

In a recursive sequence, each term after the first is represented as a function of one or more of the preceding terms. Since each term here is one fourth of its predecessor, the recursive rule of the sequence can be represented as \(a_n = \frac{a_{n-1}}{4}\) where \(a_n\) is the nth term and \(a_{n-1}\) is the previous term.

Step 3: Find the next term

The next term of the sequence can be found by applying the recursive rule to the last given term. So the next term would be \(\frac{(\frac{9}{4})}{4} = \frac{9}{16}\).

Key Concepts

Geometric Sequence

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Unlike an arithmetic sequence, which involves addition or subtraction, geometric sequences are based on multiplication and division. This makes them incredibly useful in scenarios involving exponential growth or decay.
Consider the given sequence: 144, 36, 9, \(\frac{9}{4}\), and so on. Notice how you can multiply each term by \(\frac{1}{4}\) to get to the next term. This constant factor is the hallmark of a geometric sequence, ensuring that the pattern continues indefinitely.
Key characteristics of a geometric sequence include:

• The terms grow or shrink at a constant rate.
• The ratio between any two consecutive terms is constant.

Recognizing these features makes solving and understanding them straightforward.

Common Ratio

The common ratio is the cornerstone of any geometric sequence, representing the fixed number that each term is multiplied by to get to the next one. In numerical terms, it's the factor by which a sequence's terms are stretched or compressed, influencing whether a sequence grows larger, shrinks smaller, or oscillates between values.
In our sequence example, 144, 36, 9, \(\frac{9}{4}\), the common ratio is \(\frac{1}{4}\). This means each subsequent term is one-fourth of the previous term. You can find the common ratio by dividing the second term by the first term:
\[r = \frac{36}{144} = \frac{1}{4}\]
Once you've identified this ratio:

• You can easily calculate future terms if needed.
• It helps to verify whether a sequence is arithmetic, geometric, or neither.

Knowing the common ratio is a crucial step in creating either explicit or recursive formulas for the sequence.

Recursive Sequence

A recursive sequence is a sequence in which each term is defined as a function of one or more previous terms. Unlike explicit sequences, where each term is calculated independently, recursive sequences build each term from its predecessors.
The sequence we are examining uses a recursive formula to define its terms. Our recursive formula for the sequence 144, 36, 9, \(\frac{9}{4}\) is structured as:
\[a_n = \frac{a_{n-1}}{4}\]
Here are the basic steps to formulate a recursive sequence:

• Identify the first term (also known as the initial condition), which in this example is 144.
• Establish the rule for obtaining subsequent terms. This might involve simple operations based on prior terms, as we've shown here.

Recursive sequences can model a variety of situations, especially those where each stage directly depends on the previous one, making them quite dynamic and adaptable including in economics, physics, and computer science.