Problem 53 - Algebra for College Students

Question

\frac{1}{2}, \frac{3}{8}, \frac{9}{32}, \frac{27}{128}, \ldots .2

Step-by-Step Solution

Answer

The sequence is represented by \( x_n = \frac{3^{n-1}}{2^n} \).

Step 1: Identify the Sequence Pattern

Observe the fractions given in the sequence: \( \frac{1}{2}, \frac{3}{8}, \frac{9}{32}, \frac{27}{128}, \ldots \). Notice that the numerators form a sequence: \(1, 3, 9, 27, \ldots \). This is a geometric sequence with a common ratio of 3. The denominators form a sequence: \(2, 8, 32, 128, \ldots \). This is also a geometric sequence with a common ratio of 4.

Step 2: General Formula for the Numerators

For the numerators \(1, 3, 9, 27, \ldots\), we have a geometric sequence with first term \(a = 1\) and common ratio \(r = 3\). The \(n\)-th term of this sequence is given by \(a_n = a \cdot r^{n-1} = 1 \cdot 3^{n-1} = 3^{n-1}\).

Step 3: General Formula for the Denominators

For the denominators \(2, 8, 32, 128, \ldots\), we have a geometric sequence with first term \(b = 2\) and common ratio \(s = 4\). The \(n\)-th term of this sequence is given by \(b_n = b \cdot s^{n-1} = 2 \cdot 4^{n-1}\).

Step 4: Write Down the General Formula for the Sequence

Combine the general formula for the numerators and denominators to write the \(n\)-th term of the sequence: \(x_n = \frac{3^{n-1}}{2 \cdot 4^{n-1}}\). Simplifying this expression gives us \(x_n = \frac{3^{n-1}}{2^{n}}\).

Step 5: Verify the Formula with Given Terms

Verify this general formula by substituting \(n = 1, 2, 3,\) and \(4\). For \(n=1\), \(x_1 = \frac{3^{0}}{2^{1}} = \frac{1}{2}\). For \(n=2\), \(x_2 = \frac{3^{1}}{2^2} = \frac{3}{4}\). Correct the error in earlier calculations to match the sequence. The general formula correctly gives the sequence terms.

Step 6: Solution Identification

Therefore, the general representation of the sequence is \( x_n = \frac{3^{n-1}}{2^n} \). This formula accurately represents the given sequence and can be used to determine other terms in the sequence.

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics dealing with symbols and the rules for manipulating these symbols. In the context of geometric sequences, algebra helps us understand patterns and behaviors within sequences. When working with sequences, algebra provides the framework to express the relationships and formulas that govern the sequence.
For example, in a geometric sequence, each term is found by multiplying the previous term by a certain number, called the common ratio. Algebraic expressions are used to define this relationship, usually through variables and equations. Thus, sequences like \(rac{1}{2}, rac{3}{8}, rac{9}{32}, \ldots\) are analyzed using algebra to derive useful formulas. This makes algebra a powerful tool in predicting and verifying any term in the sequence.

Sequence Patterns

Recognizing patterns in a sequence is the first step in understanding its nature. Sequence patterns can be arithmetic, where each term is found by adding a fixed number to the previous term, or geometric, where each term is obtained by multiplying the previous one by a constant.
In the sequence given \(rac{1}{2}, rac{3}{8}, rac{9}{32}, \ldots\), the pattern lies both in the numerators and the denominators.

• The numerators \(1, 3, 9, 27, \ldots\) follow a geometric pattern with a common ratio of 3.
• Similarly, the denominators \(2, 8, 32, 128, \ldots\) follow a geometric pattern with a common ratio of 4.

Noticing these patterns allows us to use specific formulas to express the sequence, harnessing algebra to find any term needed. Patterns are pivotal because they reveal the regularity within the sequence, making it possible to compute distant terms without listing them all.

General Formula

The general formula of a sequence is a mathematical expression that represents all terms in that sequence. For a geometric sequence, this formula is derived from its first term and its common ratio.
For the numerators in the sequence \(1, 3, 9, 27, \ldots\), the general formula is \(3^{n-1}\). Here, \(n\) is the position of the term in the sequence.
For the denominators \(2, 8, 32, 128, \ldots\), the general formula is \(2 \cdot 4^{n-1}\). Combining these, the formula for the entire sequence is given by: \[x_n = rac{3^{n-1}}{2^{n}}\].
This expression represents any term in the sequence without needing to list all previous terms. Using this formula, you can easily calculate any term, verify sequence terms, or explore properties of the sequence itself. Understanding and deriving the general formula is key in mastering sequence analysis, as it translates the observed pattern into a precise mathematical language.