Explicit Formula For Arithmetic Sequence And Domain Restrictions
| n | a_n |
|---|---|
| 1 | 40 |
| 2 | 47 |
| 3 | 54 |
This comprehensive guide delves into the process of deriving the explicit formula for a given arithmetic sequence. We will meticulously analyze the provided table, identify the key components of the sequence, and construct the formula. Furthermore, we will address the crucial aspect of domain restrictions, ensuring a thorough understanding of the sequence's behavior.
Understanding Arithmetic Sequences
Before diving into the specifics of our problem, let's establish a firm understanding of arithmetic sequences. An arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference. For example, the sequence 2, 5, 8, 11... is an arithmetic sequence with a common difference of 3.
The explicit formula for an arithmetic sequence provides a direct way to calculate any term in the sequence without having to know the preceding terms. This formula is generally expressed as:
a_n = a_1 + (n - 1)d
Where:
- a_n represents the nth term of the sequence.
- a_1 represents the first term of the sequence.
- n represents the term number (the position of the term in the sequence).
- d represents the common difference.
This formula essentially states that the nth term is equal to the first term plus the common difference multiplied by one less than the term number. This makes intuitive sense: to reach the nth term, you start at the first term and add the common difference (n - 1) times.
Understanding the explicit formula is crucial for working with arithmetic sequences. It allows us to:
- Calculate any term in the sequence directly.
- Identify patterns and relationships within the sequence.
- Solve problems involving arithmetic sequences, such as finding missing terms or determining the sum of a series.
Analyzing the Given Sequence
Now, let's turn our attention to the specific arithmetic sequence presented in the table:
| n | a_n |
|---|---|
| 1 | 40 |
| 2 | 47 |
| 3 | 54 |
Our first step is to identify the first term (a_1) and the common difference (d). The first term is simply the value of a_n when n is 1. From the table, we can see that a_1 = 40.
To find the common difference, we can subtract any term from its subsequent term. For instance:
- a_2 - a_1 = 47 - 40 = 7
- a_3 - a_2 = 54 - 47 = 7
This confirms that the common difference, d, is 7. The constant difference between consecutive terms is the hallmark of an arithmetic sequence, and we have successfully identified it in this case.
Having determined the first term (a_1 = 40) and the common difference (d = 7), we now possess the necessary information to construct the explicit formula for this arithmetic sequence. The next section will detail the process of substituting these values into the general formula and arriving at the specific formula for our sequence.
Constructing the Explicit Formula
With the first term (a_1 = 40) and the common difference (d = 7) firmly established, we are ready to construct the explicit formula for this arithmetic sequence. Recall the general form of the explicit formula:
a_n = a_1 + (n - 1)d
Our task is to substitute the values we have found for a_1 and d into this equation. Replacing a_1 with 40 and d with 7, we get:
a_n = 40 + (n - 1)7
This equation represents the explicit formula for the arithmetic sequence in the table. It provides a direct way to calculate the nth term (a_n) for any given term number n. To simplify the formula further, we can distribute the 7 and combine like terms:
a_n = 40 + 7n - 7
a_n = 7n + 33
Thus, the simplified explicit formula for the given arithmetic sequence is a_n = 7n + 33. This formula is equivalent to the previous one but is often preferred for its cleaner and more concise form. It clearly shows the linear relationship between the term number n and the term value a_n. For every increase of 1 in n, a_n increases by 7, reflecting the common difference.
This formula allows us to quickly calculate any term in the sequence. For example, to find the 10th term (a_10), we simply substitute n = 10 into the formula:
a_10 = 7(10) + 33 = 70 + 33 = 103
Therefore, the 10th term in the sequence is 103. This demonstrates the power of the explicit formula in providing a direct and efficient method for determining any term in the sequence. The next critical step is to consider the domain restrictions that apply to this formula, ensuring we use it correctly and avoid nonsensical results.
Domain Restrictions
Domain restrictions are an essential consideration when working with sequences and functions. They define the set of input values (n in our case) for which the formula produces valid and meaningful output values (a_n). In the context of arithmetic sequences, the domain restrictions typically stem from the nature of the sequence itself.
Arithmetic sequences are, by definition, ordered lists of numbers. Each term in the sequence occupies a specific position, and these positions are represented by positive integers. The first term is at position 1, the second term is at position 2, and so on. It is nonsensical to talk about the
