Problem 24 - 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. $$ 7,9,11,13, \dots $$

Step-by-Step Solution

Answer

The algebraic expression is \( a_n = 2n + 5 \) and the ninth term is 23.

Step 1: Identify the Pattern

Examine the sequence 7, 9, 11, 13,... This is an arithmetic sequence where each term increases by a constant value.

Step 2: Find the Common Difference

Calculate the common difference (d). In this sequence, each term increases by 2 (9 - 7 = 2, 11 - 9 = 2, 13 - 11 = 2). So, the common difference is d = 2.

Step 3: Write the General Formula

For an arithmetic sequence, the nth term is given by the formula: \( a_n = a_1 + (n-1)d \), where \(a_1\) is the first term and \(d\) is the common difference. Here, \(a_1 = 7\).

Step 4: Substitute Values into the Formula

Substitute \(a_1 = 7\) and \(d = 2\) into the general formula: \( a_n = 7 + (n-1) imes 2 \).

Step 5: Simplify the Expression

Simplify the expression for the nth term: \( a_n = 7 + 2n - 2 = 2n + 5 \). Thus, the algebraic expression is \( a_n = 2n + 5 \).

Step 6: Find the Ninth Term

To find the ninth term, substitute \(n = 9\) into the expression \( a_n = 2n + 5 \).

Step 7: Calculate the Ninth Term

Calculate \( a_9 = 2(9) + 5 = 18 + 5 = 23 \). So, the ninth term is 23.

Key Concepts

Understanding the Common Difference

In an arithmetic sequence, each term after the first is generated by adding or subtracting a fixed number. This fixed number is known as the 'common difference'. Let's consider the sequence given: 7, 9, 11, 13,...

• To find the common difference in our example, subtract any term from the following term. Let's take the first two terms: 9 - 7 = 2.
• You can check more terms (e.g., 11 - 9 = 2 and 13 - 11 = 2) to ensure it stays consistent. This proves the sequence is arithmetic.

The common difference for this particular sequence is 2, meaning each term is 2 units greater than the previous one. Recognizing this pattern is crucial for forming a general formula for the sequence. It's like finding the step size you need to move from one term to the next!

The nth Term Formula

The nth term formula for an arithmetic sequence lets you find any term's value without listing all the previous ones. Generally, it is expressed as:

• \(a_n = a_1 + (n-1)d\)

Where:

• \(a_n\) is the nth term.
• \(a_1\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the term number you are solving for.

In our example,

• the first term \(a_1\) is 7,
• and the common difference \(d\) is 2.
• By substituting these into the formula: \(a_n = 7 + (n-1) \times 2\).

This formula gives you a roadmap for finding any specific term within the sequence.

Crafting an Algebraic Expression

Algebraic expressions help in generalizing the pattern of a sequence. For our arithmetic sequence, we found the nth term formula: \(a_n = 7 + (n-1) \times 2\). From here, simplifying it will make it easier to work with.

• Expand the expression: \(a_n = 7 + 2n - 2\).
• Combine like terms: \(a_n = 2n + 5\).

The algebraic expression \(a_n = 2n + 5\) now represents this sequence in a simplified form. This expression can help you quickly find any desired term without going through every sequence step by step.

Pattern Recognition in Sequences

Pattern recognition is a powerful skill that allows you to quickly understand and predict sequences without computation. When you recognize that a sequence follows a particular pattern, like a consistent increase or decrease, it indicates an underlying structure.

In our arithmetic sequence example:

• We saw a pattern of increasing by 2, identifying it as an arithmetic sequence.
• This recognition leads us to use specific formulas, like the nth term formula, derived from arithmetic principles.

By honing your pattern recognition skills, you can easily identify these sequences and apply the correct mathematical models to solve problems efficiently.