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Magazine Chapter 6: Problem 3
In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n $$
Short Answer
Expert verified
The first five terms are 1, 2, 3, 4, 5.
Step by step solution
01
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.
02
Calculate the First Term
Substitute \(n = 1\) into the formula \(a_{n} = n\) to find the first term. So, \(a_{1} = 1\).
03
Calculate the Second Term
Substitute \(n = 2\) into the formula \(a_{n} = n\) to find the second term. Thus, \(a_{2} = 2\).
04
Calculate the Third Term
Substitute \(n = 3\) into the formula \(a_{n} = n\) to find the third term. Hence, \(a_{3} = 3\).
05
Calculate the Fourth Term
Substitute \(n = 4\) into the formula \(a_{n} = n\) to find the fourth term. As a result, \(a_{4} = 4\).
06
Calculate the Fifth Term
Substitute \(n = 5\) into the formula \(a_{n} = n\) to find the fifth term. Therefore, \(a_{5} = 5\).
07
Write the First Five Terms
The first five terms of the sequence are \(1, 2, 3, 4, 5\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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 \( a_5 = 5 \).
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鈥檚 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.
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.
