Which Of The Following Is True Regarding The Sequence Below? 7 12 , 1 6 , − 1 4 , − 2 3 \frac{7}{12}, \frac{1}{6}, -\frac{1}{4}, -\frac{2}{3} 12 7 ​ , 6 1 ​ , − 4 1 ​ , − 3 2 ​ A. The Sequence Is Arithmetic Because There Is A Common Difference Of − 5 12 -\frac{5}{12} − 12 5 ​ .B. The Sequence Is Arithmetic

by ADMIN 308 views

When it comes to sequences, understanding the different types is crucial for making accurate conclusions. In this article, we will delve into the world of arithmetic sequences and examine the given sequence to determine which of the provided statements is true.

What is an Arithmetic Sequence?

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. Arithmetic sequences can be represented in the form:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term, and 'd' is the common difference.

Examining the Given Sequence

The given sequence is:

712,16,−14,−23\frac{7}{12}, \frac{1}{6}, -\frac{1}{4}, -\frac{2}{3}

To determine if this sequence is arithmetic, we need to examine the differences between consecutive terms.

Calculating the Differences

Let's calculate the differences between consecutive terms:

  • 16−712=−112\frac{1}{6} - \frac{7}{12} = -\frac{1}{12}
  • −14−16=−112-\frac{1}{4} - \frac{1}{6} = -\frac{1}{12}
  • −23−(−14)=−512-\frac{2}{3} - (-\frac{1}{4}) = -\frac{5}{12}

As we can see, the differences between consecutive terms are not constant. The first two differences are −112-\frac{1}{12}, and the third difference is −512-\frac{5}{12}. This indicates that the sequence is not arithmetic.

Evaluating the Provided Statements

Now that we have examined the sequence and determined that it is not arithmetic, let's evaluate the provided statements:

A. The sequence is arithmetic because there is a common difference of −512-\frac{5}{12}.

This statement is false because the sequence is not arithmetic, and the common difference is not constant.

B. The sequence is arithmetic.

This statement is also false because the sequence is not arithmetic.

Conclusion

In conclusion, the given sequence is not arithmetic because the differences between consecutive terms are not constant. Therefore, neither of the provided statements is true.

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 examine the differences between consecutive terms.
  • If the differences between consecutive terms are not constant, the sequence is not arithmetic.

In our previous article, we explored the concept of arithmetic sequences and examined a given sequence to determine if it was arithmetic. In this article, we will address some frequently asked questions about arithmetic sequences.

Q: What is the difference between an arithmetic sequence and a geometric sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric sequence, on the other hand, is a sequence of numbers in which the ratio between any two consecutive terms is constant.

Q: How do I determine if a sequence is arithmetic?

A: To determine if a sequence is arithmetic, you need to examine the differences between consecutive terms. If the differences between consecutive terms are constant, then the sequence is arithmetic.

Q: What is the formula for an arithmetic sequence?

A: The formula for an arithmetic sequence is:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term, and 'd' is the common difference.

Q: How do I find the common difference in an arithmetic sequence?

A: To find the common difference in an arithmetic sequence, you can subtract any term from the previous term. For example, if the sequence is:

2, 5, 8, 11, ...

You can find the common difference by subtracting the first term from the second term:

5 - 2 = 3

Q: Can an arithmetic sequence have a negative common difference?

A: Yes, an arithmetic sequence can have a negative common difference. For example:

3, 1, -1, -3, ...

In this sequence, the common difference is -2.

Q: Can an arithmetic sequence have a zero common difference?

A: Yes, an arithmetic sequence can have a zero common difference. For example:

1, 1, 1, 1, ...

In this sequence, the common difference is 0.

Q: What is the sum of an arithmetic sequence?

A: The sum of an arithmetic sequence can be found using the formula:

S = n/2 * (a + l)

where 'S' is the sum, 'n' is the number of terms, 'a' is the first term, and 'l' is the last term.

Q: What is the average of an arithmetic sequence?

A: The average of an arithmetic sequence is the same as the median, which is the middle term. If the sequence has an even number of terms, the average is the average of the two middle terms.

Q: Can an arithmetic sequence have a fractional common difference?

A: Yes, an arithmetic sequence can have a fractional common difference. For example:

1/2, 1/4, 1/8, 1/16, ...

In this sequence, the common difference is -1/4.

Conclusion

Arithmetic sequences are an important concept in mathematics, and understanding them can help you solve a wide range of problems. By answering these frequently asked questions, we hope to have provided you with a better understanding of arithmetic sequences and how to work with them.

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, you need to examine the differences between consecutive terms.
  • The formula for an arithmetic sequence is a, a + d, a + 2d, a + 3d, ...
  • The common difference in an arithmetic sequence can be found by subtracting any term from the previous term.
  • An arithmetic sequence can have a negative, zero, or fractional common difference.