Problem 15 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=\frac{1}{2}, r=\frac{2}{3} $$

Step-by-Step Solution

Answer

The explicit formula for the geometric sequence is \(a_n = \frac{1}{2} * (\frac{2}{3})^{(n-1)}\). The first five terms of the sequence are \(\frac{1}{2}, \frac{1}{3}, \frac{2}{9}, \frac{4}{27}, \frac{8}{81}\)

Step 1: Write the Explicit Formula

Plug the given values of \(a_1\) and \(r\) into the general formula for a geometric sequence. Thus, the formula for this sequence is \(a_n = \frac{1}{2} * (\frac{2}{3})^{(n-1)}\)

Step 2: Generate the First Five Terms

Substitute \(n = 1, 2, 3, 4, 5\) into the formula and calculate the results: \(a_1 = \frac{1}{2} * (\frac{2}{3})^{(1-1)} = \frac{1}{2}\), \(a_2 = \frac{1}{2} * (\frac{2}{3})^{(2-1)} = \frac{1}{3}\), \(a_3 = \frac{1}{2} * (\frac{2}{3})^{(3-1)} = \frac{2}{9}\), \(a_4 = \frac{1}{2} * (\frac{2}{3})^{(4-1)} = \frac{4}{27}\), \(a_5 = \frac{1}{2} * (\frac{2}{3})^{(5-1)} = \frac{8}{81}\)

Key Concepts

Explicit Formula in Geometric Sequences

In the realm of sequences, the explicit formula is your best friend when it comes to easily generating terms. Unlike recursive formulas, which rely on the knowledge of the previous term, the explicit formula provides a direct way to find any term in a sequence without having to know the value of the previous one.

For a geometric sequence, the explicit formula is written as:

• \( a_n = a_1 \times r^{(n-1)} \)

Here, \( a_n \) stands for the nth term of the sequence, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) represents the position of the term in the sequence.
This formula is particularly powerful and efficient when you need to calculate terms that are far apart in the sequence, without recounting every single term before them.
In our example, by plugging \( a_1 = \frac{1}{2} \) and \( r = \frac{2}{3} \) into this formula, we readily obtain the specific formula for our sequence: \( a_n = \frac{1}{2} \times \left(\frac{2}{3}\right)^{(n-1)} \). This leads us straight to each term with ease.

Understanding Sequence Terms

Sequence terms are the individual elements or numbers in a sequence that follow a certain order and pattern dictated by the sequence's rule. In a geometric sequence, each term is generated by multiplying the previous term by a fixed, non-zero number known as the common ratio.

The first term is critical because it is the building block from which the rest of the sequence emerges.
In our specific sequence example, we start with the first term \( a_1 = \frac{1}{2} \). By consistently multiplying by the common ratio \( r = \frac{2}{3} \), we get the following sequence terms:

• \( a_1 = \frac{1}{2} \)
• \( a_2 = \frac{1}{3} \)
• \( a_3 = \frac{2}{9} \)
• \( a_4 = \frac{4}{27} \)
• \( a_5 = \frac{8}{81} \)

These numbers build upon each other, yet each can be independently calculated through the explicit formula as demonstrated.
This consistency and predictability make sequence terms highly useful in various mathematical and real-world applications.

The Role of Mathematical Formulae

Mathematical formulae are essentially like recipes in mathematics; they provide a set of instructions on how to calculate or compute something. They package complex operations into understandable and usable equations.

For geometric sequences, the explicit formula is the mathematical tool used to define and generate sequence terms efficiently.
This formula takes the abstract concept of multiplication and exponents and applies it directly to creating practical numeric terms within a sequence.
The power and elegance of a mathematical formula lie in its simplicity and ability to generalize a solution to an entire class of problems. In our scenario, the explicit formula \( a_n = \frac{1}{2} \times \left(\frac{2}{3}\right)^{(n-1)} \) serves as a blueprint that anyone can use to find the desired term of this geometric sequence.
It's incredible how such a compact notation can uncover the entire future of the sequence’s progression. Understanding and manipulating these formulae are crucial skills in mathematical problem-solving and analysis.