Problem 4 - Algebra for College Students

Question

Find the general term of the sequence \(5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots .\) \(5(2)^{1-n}\)

Step-by-Step Solution

Answer

The general term is \(5 \times 2^{1-n}\).

Step 1: Identify the sequence pattern

By examining the sequence, we have the terms: 5, \(\frac{5}{2}\), \(\frac{5}{4}\), \(\frac{5}{8}\), etc. Notice that each term can be written as \(5 \times \frac{1}{2^{n-1}}\), where \(n\) represents the position of each term.

Step 2: Write the general term

Given the pattern identified in Step 1, the general term of the sequence can be expressed as \(a_n = 5 \times \left(\frac{1}{2^{n-1}}\right)\). This can be further simplified using properties of exponents.

Step 3: Simplify the expression

The expression \(a_n = 5 \times \left(\frac{1}{2^{n-1}}\right)\) can be rewritten by using the negative exponent: \(a_n = 5 \times 2^{-(n-1)} = 5 \times 2^{1-n}\).

Step 4: Verify the expression

To ensure correctness, substitute \(n = 1, 2, 3, 4\) into the formula \(5 \times 2^{1-n}\) to confirm it produces the terms 5, \(\frac{5}{2}\), \(\frac{5}{4}\), \(\frac{5}{8}\) respectively.

Key Concepts

General Term of Sequence

In arithmetic and geometric sequences, the general term is an expression that defines each term in the sequence in relation to its position. This expression is essential because it allows you to find any term of the sequence without having to list all preceding terms.
The general term for a sequence is often written as \(a_n\), where \(n\) signifies the term number. For the sequence \(5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}, \ldots\), we identified the general term as \(a_n = 5 \times \left(\frac{1}{2^{n-1}}\right)\).
This showcases how the sequence decreases by multiplying by \(\frac{1}{2}\) as the number \(n\) increases. Simplifying this further using exponent properties, the term becomes \(a_n = 5 \times 2^{1-n}\).
With this general formula, you can determine any term in the sequence by simply plugging in the position number \(n\). This method not only saves time but also enhances understanding of how sequences evolve.

Sequence Pattern Identification

Recognizing the pattern in a sequence is crucial for determining its general term. In our sequence, the pattern reveals itself through consistent changes between consecutive terms.
By examining \(5, \frac{5}{2}, \frac{5}{4}, \frac{5}{8}\), you notice that each term is exactly half of the previous one when excluding the consistent factor of 5.

• The first term \(5 = 5 \times 1\)
• The second term \(\frac{5}{2} = 5 \times \frac{1}{2}\)
• The third term \(\frac{5}{4} = 5 \times \frac{1}{4}\ = 5 \times \left(\frac{1}{2^2}\right)\)
• And so on.

Each term can be seen as \(5\) multiplied by \(\frac{1}{2^{n-1}}\). Recognizing this pattern allows us to construct a general term which provides a compact way to express the entire sequence.

Properties of Exponents

Understanding the properties of exponents can simplify many expressions, including those related to sequences. The properties are mathematical rules that allow me to manipulate expressions for easier calculations or representations.
In our sequence, the expression \(5 \times \frac{1}{2^{n-1}}\) can be rewritten using the exponent property of negative exponents: \(\frac{1}{2^{n-1}} = 2^{-(n-1)}\).
This results in a simpler form \(a_n = 5 \times 2^{1-n}\). This not only streamlines our calculations but provides deeper insights into the nature of the sequence's progression.

• When multiplying powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• With a negative exponent, it translates to reciprocal: \(a^{-n} = \frac{1}{a^n}\).

Mastery of these properties allows you to work with complex sequences and equations efficiently, ultimately boosting your overall math prowess.