Problem 19 - 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. $$ 2,4,6,8, \dots $$

Step-by-Step Solution

Answer

The expression for the sequence is \(a_n = 2n\), and the ninth term is 18.

Step 1: Identify the Pattern in the Sequence

The given sequence is \(2, 4, 6, 8, \ldots\). This sequence consists of terms where each term increases by 2 from the previous term. This suggests that it is an arithmetic sequence with a common difference \(d = 2\).

Step 2: Formulate the General Expression for the Sequence

For an arithmetic sequence, the \(n\)-th term \(a_n\) can be expressed using the formula \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. For this sequence, \(a_1 = 2\) and \(d = 2\). Thus, the expression for the \(n\)-th term is \(a_n = 2 + (n-1) \cdot 2\).

Step 3: Simplify the General Expression

Simplify the expression \(a_n = 2 + (n-1) \cdot 2\) to get \(a_n = 2 + 2n - 2\). This further simplifies to \(a_n = 2n\). Thus, the algebraic expression that represents \(a_n\) for this sequence is \(a_n = 2n\).

Step 4: Calculate the Ninth Term

To find the ninth term, substitute \(n = 9\) into the expression \(a_n = 2n\). This gives \(a_9 = 2 \times 9 = 18\). Thus, the ninth term of the sequence is 18.

Key Concepts

Understanding Algebraic Expressions

In mathematics, algebraic expressions are combinations of numbers, variables, and arithmetic operations. They play a crucial role in defining sequences, functions, and various mathematical concepts. When creating an algebraic expression for an arithmetic sequence, the aim is to find a formula that can efficiently represent any term in the sequence.
This formula often involves a pattern or relationship among the terms, which can be captured using variables.

• For example, in an arithmetic sequence, variables help express the relationship between a term and its position in the sequence.
• Combined with constants and operations, these expressions simplify calculations by representing complex sequences with a simple formula.

To establish such expressions, one first identifies a pattern, such as a consistent increase or decrease between consecutive terms. With the pattern recognized, the expression can then include variables to represent the position of each term, providing a compact and powerful tool in analyzing sequences.

Identifying the Common Difference

In an arithmetic sequence, one of the key elements is the "common difference." This term refers to the constant amount by which consecutive terms in the sequence change. Recognizing and calculating this difference is a fundamental step in analyzing or representing arithmetic sequences.
The common difference is usually denoted by the letter \( d \). It is calculated by subtracting any term in the sequence from the term that follows it.

• For the sequence \(2, 4, 6, 8, \ldots\), the common difference \( d \) is 2 because each successive number increases by 2.
• Knowing the common difference helps formulate the expression for the sequence's terms.

Once determined, the common difference allows students to predict subsequent terms in the sequence and understand the pattern's behavior. It's like the heartbeat of the sequence, dictating its rhythm and progression. With \( d \), the student can easily manipulate and identify specific positions within the sequence.

The Nth Term Formula Explained

The concept of an "nth term formula" is an essential part in arithmetic sequences, allowing you to directly find any term's value without sequentially counting from the beginning. This formula is expressed as \( a_n = a_1 + (n-1)d \).
Let's break it down. Each element in this formula has a particular role:

• \( a_1 \) represents the first term in the sequence. In our example sequence, \( a_1 = 2 \).
• \( n \) is the term's position number in the sequence.
• \( d \) is the common difference, which is 2 for our sequence.

When you plug these into the formula, you get \( a_n = 2 + (n-1) imes 2 \). Simplifying yields the expression \( a_n = 2n \). This means, for any term position \( n \), you can find the value quickly by multiplying \( n \) by 2.
This simplified algebraic expression empowers students to calculate any term of the sequence swiftly, thereby mastering the sequence's nature in a few steps. Using the formula effectively minimizes error by providing a systematic approach to finding terms regardless of sequence length, cementing understanding in practical calculation.