Problem 26 - Algebra 2 and Trigonometry

Question

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

Step-by-Step Solution

Answer

The expression is \(a_{n} = \frac{1}{n+1}\), and the ninth term is \(\frac{1}{10}\).

Step 1: Identify the Pattern

Examine the sequence: \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots \). Notice that the denominators are consecutive integers starting from 2. This pattern helps us to develop a general expression for the nth term.

Step 2: Develop the Expression for the nth Term

From the pattern, observe that the denominator of each term is \(n+1\) (where \(n\) is the term number starting from 1). Thus, the expression for the nth term is \(a_{n} = \frac{1}{n+1}\).

Step 3: Calculate the Ninth Term

To find the ninth term, substitute \(n=9\) into the expression \(a_{n} = \frac{1}{n+1}\). Therefore, \(a_{9} = \frac{1}{9+1} = \frac{1}{10}\).

Key Concepts

Sequences

In mathematics, sequences are ordered lists of numbers that often follow a specific pattern or rule. This is a fundamental concept in algebra since it allows us to study numerical patterns and predict future values based on a given formula. A sequence can be finite or infinite.
This concept is important because by understanding how sequences work, you can easily move on to handling more complex algebraic expressions.
For sequences that are part of a pattern, each specific number in the list is known as a term. Each term's position in the sequence plays a crucial role, as it often helps in identifying the rule which governs the sequence. In our exercise, you are given terms like \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), and so on. Recognizing these numbers as part of a sequence is the first step towards finding a relationship among them.

nth Term

The concept of the "nth term" refers to a generic formula that allows you to calculate the value of any term within a sequence, based on its position.
In algebra, determining the nth term is crucial because it simplifies the process of determining specific terms without having to list all preceding ones.For example, in our given sequence, the rule revolves around the denominators, which increase incrementally. By identifying this, we set up an expression for any term in the sequence using "n" as a variable to represent the position of the term.
Using the pattern derived, we know the nth term expression is \(a_{n} = \frac{1}{n+1}\). This expression is derived by observing that each term’s denominator increases by 1 with each step, starting from 2. By applying this expression, you can figure out where any term falls in the sequence without listing all the terms up to that point.

Pattern Recognition

Pattern recognition is a skill that involves identifying trends, regularities, or structures within data. In the context of sequences, it is about noticing how numbers relate to each other.
This skill is essential in developing a formula for the nth term and analyzing how sequences progress.In algebra, recognizing patterns is often your first move. For the sequence \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\), and onwards, recognizing that only the denominators progress (and increment by 1) is key.
When you accurately identify a pattern, you can establish a formula to find any term. For this sequence, the pattern was that every next term’s denominator was an incremented integer, starting from 2. This guidance helped develop the general expression \(a_{n} = \frac{1}{n+1}\) for each nth term.

Expression Development

Expression development is the process of constructing an algebraic expression to represent any term within a sequence. It involves translating the recognized sequence pattern into mathematical language.
This step is like putting the pieces of a puzzle together in math. Once a pattern is identified, expression development utilizes algebraic operations to encapsulate that pattern into a usable formula. In our sequence, the recognized pattern highlighted how the denominators behaved, increasing consecutively. By translating this into an equation, we find that the formula is \(a_{n} = \frac{1}{n+1}\). The role of expression development is pivotal because it turns observation into utility. It allows you to predict any arbitrary term in the sequence, such as finding \(a_9\). Using the developed expression, substituting \(n = 9\) gave us an answer of \(\frac{1}{10}\). Thus, expression development empowers you to explore deeper mathematical queries with ease.