Problem 17 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots $$

Step-by-Step Solution

Answer

The recursive formula for the sequence is \(a_n = \frac{1}{2} * a_{n-1}\) for \(n>1\) and the first term \(a_1 = \frac{1}{2}\). The next term after \( \frac{1}{32}\) is \( \frac{1}{64}\).

Step 1: Identify the sequence pattern

First, look at the presented sequence: \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots \). Noticing the sequence's pattern, every term seems to be a half of the term before it. So this sequence can be described by a recursive formula.

Step 2: Formulating the recursive formula

A recursive formula always has two parts: the starter, the first term (or few terms) that get the process going, and the recursive pattern, how the sequence's previous term(s) can be manipulated to get the next term. In this case, the first term \(a_1 = \frac{1}{2}\) and every next term can be found by halving (multiplying by \( \frac{1}{2}\)) the preceding term. This leads us to the recursive formula: \(a_n = \frac{1}{2} * a_{n-1}\) where \(n>1\).

Step 3: Finding the next term

Now, to find the next term after \( \frac{1}{32}\), we apply the recursive formula. Thus, the next term \(a_6 = \frac{1}{2}*\frac{1}{32} = \frac{1}{64}\).

Key Concepts

Geometric Sequences

Geometric sequences are fascinating mathematical constructs that capture the essence of progression. In a geometric sequence, each term is derived by multiplying the previous one by a constant called the common ratio. This repeated multiplication quickly demonstrates growth or decay. These sequences are prevalent in various fields like finance for compound interest calculations and even in nature, illustrating growth patterns in various phenomena.
To develop a deeper understanding, consider the simple sequence: \(2, 4, 8, 16, 32, \dots\). Here, the common ratio is 2. You get each subsequent term by multiplying the previous term by 2. Symbolically, if the first term \(a_1 = 2\), the nth term \(a_n\) is obtained using the formula \(a_n = a_1 \times r^{(n-1)}\), where \(r\) is the common ratio.
In the sequence from the original exercise, the common ratio is \(\frac{1}{2}\). Understanding geometric sequences helps in identifying growth patterns, making predictions, and solving problems related to exponential growth and decay.

Fraction Sequences

Fraction sequences often involve terms that are fractions, and they can exhibit fascinating patterns based on multiplication or division. The sequence \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots\) is a prime example to illustrate such sequences.
In the given sequence, each term is a fraction that results from multiplying the previous term by another fraction - specifically \(\frac{1}{2}\). It's important to note that shrinking fractions like these are a hallmark of approaches where quantities reduce over time or with processes that involve splitting.
Understanding how to manipulate fractions in sequences relies on taking note of the operation (multiplication or division) applied regularly. Here, the concept of multiplying by \(\frac{1}{2}\) is instrumental, as seen in the pattern that shrinks the denominator in powers of 2. Such sequences are particularly helpful in understanding problems involving halves, similar to those seen in probability or resource allocation issues.

Sequence Pattern Identification

Sequence pattern identification is an essential skill in mathematics, allowing you to decipher any sequence's underlying rule or formula quickly. This process begins with examining the starting points and transitions between numbers. For instance, in the sequence \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \cdots\), identifying that the terms halve as they progress helps in formulating a recursive relationship.
The main steps are:

• Observation: Start by observing changes between consecutive terms.
• Pattern Recognition: Look for consistent operations applied, such as addition, multiplication, or division.
• Formulation: Express this identified pattern mathematically. For recursive sequences, you'll typically express the nth term based on the previous term.

Recognizing patterns equips you with analytical tools necessary to solve sequences and extrapolate future values logically. In the provided exercise, the pattern of multiplying by \(\frac{1}{2}\) was crucial for determining both the recursive formula and finding subsequent terms in fraction sequences.