Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n $$

Step-by-Step Solution

Answer

The first five terms are 1, 2, 3, 4, 5.

Step 1: Understand the Formula

The sequence is defined by the formula \(a_{n} = n\). This means each term is exactly equal to its position number in the sequence. For example, the first term \(a_{1}\) will be 1, the second term \(a_{2}\) will be 2, and so on.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the formula \(a_{n} = n\) to find the first term. So, \(a_{1} = 1\).

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the formula \(a_{n} = n\) to find the second term. Thus, \(a_{2} = 2\).

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the formula \(a_{n} = n\) to find the third term. Hence, \(a_{3} = 3\).

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the formula \(a_{n} = n\) to find the fourth term. As a result, \(a_{4} = 4\).

Step 6: Calculate the Fifth Term

Substitute \(n = 5\) into the formula \(a_{n} = n\) to find the fifth term. Therefore, \(a_{5} = 5\).

Step 7: Write the First Five Terms

The first five terms of the sequence are \(1, 2, 3, 4, 5\).

Key Concepts

Sequence Formula

When dealing with sequences, we often rely on what is called a 'sequence formula'. This formula is crucial as it defines how each term in the sequence is generated. In the context of arithmetic sequences, each term is typically calculated by adding a constant difference to the previous term. However, the sequence in our example has a particularly straightforward formula:

• The given formula is \( a_{n} = n \).
• This formula indicates that each term \( a_{n} \) is equal to its position number in the sequence.
• This makes it a simple linear function and not typically classified as an arithmetic sequence because it doesn't have a constant difference between terms.

To understand any formula thoroughly, it's important to identify such key elements as the manner in which each term relates to its position. This will aid in better comprehension of the sequence's structure.

Term Calculation

Calculating terms in a sequence involves substituting values into the sequence formula to find specific terms. This sequence where \( a_n = n \) makes this particularly straightforward.

• For the first term, substitute \( n = 1 \) to get \( a_1 = 1 \).
• For the second term, substitute \( n = 2 \), resulting in \( a_2 = 2 \).
• Similarly, for \( n = 3, 4, \) and \( 5 \), you get terms \( a_3 = 3, a_4 = 4, \) and \( a_5 = 5 \).

This process is straightforward for this specific formula because it is an identity function, meaning each term equals its input value. Term calculation generally becomes more complex with more intricate formulas, requiring understanding of operations within the formula.

Position Number

In sequences, the position number, denoted by \( n \) in the formula \( a_n = n \), plays a critical role. It represents the place of a term in the sequence. Here’s why the position number matters:

• It tells us which number to substitute into the formula to find the specific term.
• It acts as both the input and output in this particular sequence since each term equals its position.
• The concept of position number helps in organizing and predicting the terms systematically.

Even in more complicated sequences, understanding the position number helps locate which term you're solving for and assists in ensuring calculations are done for the right term. It is foundational in sequence study.

Mathematics Education

Mathematics education covers a multitude of concepts, from simple arithmetic to complex calculus. Understanding sequences is one key aspect that enhances logical reasoning and pattern recognition. Here's why studying sequences is beneficial:

• They help develop the ability to discern patterns and predict outcomes.
• Sequences, such as geometric and arithmetic types, form the base for understanding more advanced mathematical concepts.
• Through sequences like the one defined by \( a_n = n \), students learn about direct proportionality and linear relationships.

As learners engage with sequences, they gain skills in recognizing regularities and differences, key to mastering further mathematical challenges. Such foundational concepts foster a solid mathematical mindset, preparing students for more complex studies and problem-solving in real-world scenarios.