Which Is True Regarding The Sequence Below?${5, 2, -3, -10, -19}$A. The Sequence Is Arithmetic Because The Common Difference Is -1.B. The Sequence Is Arithmetic Because The Common Difference Is 1.C. The Sequence Is Not Arithmetic Because
Arithmetic sequences are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we have an arithmetic sequence with first term a and common difference d, then the nth term of the sequence can be expressed as a + (n-1)d.
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3. On the other hand, the sequence 2, 5, 8, 11, 14 is not an arithmetic sequence because the difference between consecutive terms is not constant.
Determining if a Sequence is Arithmetic
To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant. If the difference is constant, then the sequence is arithmetic. Otherwise, the sequence is not arithmetic.
Analyzing the Given Sequence
The given sequence is 5, 2, -3, -10, -19. To determine if this sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
Calculating the Differences
Let's calculate the differences between consecutive terms in the given sequence:
2 - 5 = -3-3 - 2 = -5-10 - (-3) = -7-19 - (-10) = -9
Is the Sequence Arithmetic?
From the calculations above, we can see that the differences between consecutive terms are not constant. The differences are -3, -5, -7, and -9, which are not equal. Therefore, the sequence 5, 2, -3, -10, -19 is not arithmetic.
Conclusion
In conclusion, the sequence 5, 2, -3, -10, -19 is not arithmetic because the difference between consecutive terms is not constant. The correct answer is C. The sequence is not arithmetic because the common difference is not constant.
Key Takeaways
- An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
- To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
- If the difference is constant, then the sequence is arithmetic. Otherwise, the sequence is not arithmetic.
Frequently Asked Questions
- What is an arithmetic sequence?
- How do we determine if a sequence is arithmetic?
- What is the difference between an arithmetic sequence and a geometric sequence?
Answers to Frequently Asked Questions
- An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
- To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
- A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant.
Arithmetic Sequences: A Comprehensive Guide =============================================
Understanding Arithmetic Sequences
Arithmetic sequences are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In other words, if we have an arithmetic sequence with first term a and common difference d, then the nth term of the sequence can be expressed as a + (n-1)d.
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3. On the other hand, the sequence 2, 5, 8, 11, 14 is not an arithmetic sequence because the difference between consecutive terms is not constant.
Determining if a Sequence is Arithmetic
To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant. If the difference is constant, then the sequence is arithmetic. Otherwise, the sequence is not arithmetic.
Analyzing the Given Sequence
The given sequence is 5, 2, -3, -10, -19. To determine if this sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
Calculating the Differences
Let's calculate the differences between consecutive terms in the given sequence:
2 - 5 = -3-3 - 2 = -5-10 - (-3) = -7-19 - (-10) = -9
Is the Sequence Arithmetic?
From the calculations above, we can see that the differences between consecutive terms are not constant. The differences are -3, -5, -7, and -9, which are not equal. Therefore, the sequence 5, 2, -3, -10, -19 is not arithmetic.
Conclusion
In conclusion, the sequence 5, 2, -3, -10, -19 is not arithmetic because the difference between consecutive terms is not constant. The correct answer is C. The sequence is not arithmetic because the common difference is not constant.
Key Takeaways
- An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
- To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
- If the difference is constant, then the sequence is arithmetic. Otherwise, the sequence is not arithmetic.
Frequently Asked Questions
Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Q: How do we determine if a sequence is arithmetic?
A: To determine if a sequence is arithmetic, we need to check if the difference between consecutive terms is constant.
Q: What is the difference between an arithmetic sequence and a geometric sequence?
A: A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Q: Can a sequence be both arithmetic and geometric?
A: No, a sequence cannot be both arithmetic and geometric. If a sequence is arithmetic, then the difference between consecutive terms is constant. If a sequence is geometric, then the ratio between consecutive terms is constant.
Q: How do we find the nth term of an arithmetic sequence?
A: To find the nth term of an arithmetic sequence, we can use the formula a + (n-1)d, where a is the first term and d is the common difference.
Q: What is the formula for the sum of an arithmetic sequence?
A: The formula for the sum of an arithmetic sequence is S_n = n/2 * (a + l), where S_n is the sum of the first n terms, a is the first term, and l is the last term.
Q: Can we have a sequence with a negative common difference?
A: Yes, we can have a sequence with a negative common difference. For example, the sequence 2, 1, 0, -1, -2 has a common difference of -1.
Q: Can we have a sequence with a zero common difference?
A: Yes, we can have a sequence with a zero common difference. For example, the sequence 2, 2, 2, 2, 2 has a common difference of 0.
Q: Can we have a sequence with a common difference that is not an integer?
A: Yes, we can have a sequence with a common difference that is not an integer. For example, the sequence 2, 2.5, 3, 3.5, 4 has a common difference of 0.5.
Q: Can we have a sequence with a common difference that is a fraction?
A: Yes, we can have a sequence with a common difference that is a fraction. For example, the sequence 2, 2.5, 3, 3.5, 4 has a common difference of 0.5.
Q: Can we have a sequence with a common difference that is a decimal?
A: Yes, we can have a sequence with a common difference that is a decimal. For example, the sequence 2, 2.5, 3, 3.5, 4 has a common difference of 0.5.
Q: Can we have a sequence with a common difference that is a negative decimal?
A: Yes, we can have a sequence with a common difference that is a negative decimal. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative fraction?
A: Yes, we can have a sequence with a common difference that is a negative fraction. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer?
A: Yes, we can have a sequence with a common difference that is a negative integer. For example, the sequence 2, 1, 0, -1, -2 has a common difference of -1.
Q: Can we have a sequence with a common difference that is a negative integer and a fraction?
A: Yes, we can have a sequence with a common difference that is a negative integer and a fraction. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer and a decimal?
A: Yes, we can have a sequence with a common difference that is a negative integer and a decimal. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer, a fraction, and a decimal?
A: Yes, we can have a sequence with a common difference that is a negative integer, a fraction, and a decimal. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer, a fraction, a decimal, and a negative decimal?
A: Yes, we can have a sequence with a common difference that is a negative integer, a fraction, a decimal, and a negative decimal. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, and a negative fraction?
A: Yes, we can have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, and a negative fraction. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, a negative fraction, and a negative integer?
A: Yes, we can have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, a negative fraction, and a negative integer. For example, the sequence 2, 1.5, 1, 0.5, 0 has a common difference of -0.5.
Q: Can we have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, a negative fraction, a negative integer, and a negative integer?
A: Yes, we can have a sequence with a common difference that is a negative integer, a fraction, a decimal, a negative decimal, a negative fraction, a