An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted as d. In simpler terms, you obtain the next term in the sequence by adding or subtracting the same value from the previous term. Recognizing arithmetic sequences is a fundamental skill in mathematics, especially in algebra and calculus. Understanding these sequences helps in predicting patterns, solving problems related to progressions, and grasping more advanced mathematical concepts.
To identify an arithmetic sequence, we calculate the difference between consecutive terms. If this difference remains the same throughout the sequence, then it is indeed an arithmetic sequence. For example, the sequence 2, 4, 6, 8,... is an arithmetic sequence because the common difference is 2 (4-2 = 2, 6-4 = 2, 8-6 = 2). On the other hand, a sequence like 1, 2, 4, 8,... is not an arithmetic sequence because the differences between consecutive terms are not constant (2-1 = 1, 4-2 = 2, 8-4 = 4).
The general form of an arithmetic sequence can be represented as: a, a + d, a + 2d, a + 3d, ..., where 'a' is the first term and 'd' is the common difference. This representation helps in understanding the structure of arithmetic sequences and in deriving formulas for the nth term and the sum of the first n terms. The nth term of an arithmetic sequence can be found using the formula: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference. This formula is crucial for finding any term in the sequence without having to list all the preceding terms.
The sum of the first n terms of an arithmetic sequence can be calculated using the formula: S_n = n/2 * [2a + (n-1)d], where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference. Alternatively, if the first term (a) and the last term (l) are known, the sum can also be found using the formula: S_n = n/2 * (a + l). These formulas are essential for solving various problems related to arithmetic series, such as finding the total number of seats in a stadium or calculating the total savings over a period with consistent increments.
Arithmetic sequences are not just abstract mathematical constructs; they have numerous real-world applications. They appear in simple scenarios, such as calculating monthly salary increases, determining the number of seats in rows of a theater, and modeling uniformly accelerating objects in physics. They are also used in more complex applications, such as financial planning, computer programming, and statistical analysis. Recognizing and understanding arithmetic sequences is therefore beneficial in a wide range of practical contexts, making it a valuable skill for students and professionals alike.
To determine which of the given sequences is an arithmetic sequence, we need to examine the differences between consecutive terms in each sequence. An arithmetic sequence has a constant difference between successive terms. We will calculate these differences for each option to identify the one that exhibits this constant difference. This process involves subtracting each term from the term that follows it and checking for consistency across the sequence. Let's start by analyzing each option individually to identify the common difference, if it exists.
Option A: $-5, -7, -10, -14, -19, \ldots$
To check if this sequence is arithmetic, we calculate the differences between consecutive terms:
- -7 - (-5) = -2
- -10 - (-7) = -3
- -14 - (-10) = -4
- -19 - (-14) = -5
Since the differences (-2, -3, -4, -5) are not constant, this sequence is not an arithmetic sequence. The differences between the terms are changing, indicating that there is no common difference. Therefore, option A can be eliminated as it does not meet the criteria for an arithmetic sequence. This variability in differences suggests that the sequence follows a different pattern, possibly a quadratic or exponential relationship, rather than a linear progression characteristic of arithmetic sequences.
Option B: $1.5, -1.5, 1.5, -1.5, \ldots$
We calculate the differences between consecutive terms:
- -1.5 - 1.5 = -3
-
- 5 - (-1.5) = 3
- -1.5 - 1.5 = -3
The differences alternate between -3 and 3, which means the difference is not constant. This sequence is not an arithmetic sequence. Instead, this sequence is an example of an oscillating sequence, where the terms alternate between two values. Such sequences do not have a common difference and thus do not fit the definition of an arithmetic sequence. Recognizing these patterns is important for differentiating between various types of sequences.
Option C: $4.1, 5.1, 6.2, 7.2, \ldots$
Let's calculate the differences between consecutive terms:
-
- 1 - 4.1 = 1.0
-
- 2 - 5.1 = 1.1
-
- 2 - 6.2 = 1.0
The differences (1.0, 1.1, 1.0) are not constant. Therefore, this sequence is not an arithmetic sequence. The slight variations in the differences indicate that while the sequence might appear to have a consistent increase, it does not adhere to the strict definition of an arithmetic sequence where the difference must be exactly the same between each pair of consecutive terms. This highlights the importance of precise calculations when identifying arithmetic sequences.
Option D: $-1.5, -1, -0.5, 0, \ldots$
Calculating the differences between consecutive terms:
- -1 - (-1.5) = 0.5
- -0.5 - (-1) = 0.5
- 0 - (-0.5) = 0.5
The difference between consecutive terms is consistently 0.5. This constant difference indicates that this sequence is indeed an arithmetic sequence. The common difference d is 0.5. This sequence represents a linear progression where each term is obtained by adding 0.5 to the previous term. This makes option D the correct choice as it satisfies the definition of an arithmetic sequence.
After analyzing each of the given sequences, we can definitively identify which one represents an arithmetic sequence. By calculating the differences between consecutive terms, we looked for a consistent, or common, difference. Options A, B, and C did not exhibit a constant difference, meaning they were not arithmetic sequences. However, Option D, $-1.5, -1, -0.5, 0, \ldots$, showed a consistent difference of 0.5 between each term. Therefore, this is the correct arithmetic sequence among the choices.
The key to identifying arithmetic sequences lies in the constant difference between consecutive terms. This difference, known as the common difference, is the defining characteristic of arithmetic sequences. Sequences that do not have this constant difference follow different patterns, such as geometric, quadratic, or other types of progressions. Understanding this principle allows for the quick and accurate identification of arithmetic sequences in various mathematical contexts.
In conclusion, the correct answer is Option D, as it is the only sequence with a constant difference between its terms, thus fitting the definition of an arithmetic sequence. This exercise demonstrates the importance of carefully examining sequences and applying the fundamental definition of an arithmetic sequence to correctly identify them.
Correct Answer: D. $-1.5, -1, -0.5, 0, \ldots$
