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

Mar 13, 2025 by ADMIN 306 views

Which of the Following is True Regarding the Sequence Below?

The given sequence is $\frac{7}{12}, \frac{1}{6},-\frac{1}{4},-\frac{2}{3}$ . In this article, we will analyze the sequence and determine whether it is arithmetic or not.

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 called the common difference. For example, the sequence $2, 4, 6, 8, 10$ is an arithmetic sequence because the common difference is $2$ .

Is the Given Sequence Arithmetic?

To determine whether the given sequence is arithmetic, we need to find the common difference between any two consecutive terms. Let's calculate the differences between the consecutive terms:

$\frac{1}{6} - \frac{7}{12} = -\frac{1}{12}$
$-\frac{1}{4} - \frac{1}{6} = -\frac{1}{12}$
$-\frac{2}{3} - (-\frac{1}{4}) = -\frac{5}{12}$

As we can see, the differences between the consecutive terms are not constant. The common difference is not the same throughout the sequence. Therefore, the given sequence is not arithmetic.

Why is the Sequence Not Arithmetic?

The sequence is not arithmetic because the common difference is not constant. The common difference is $-\frac{1}{12}$ between the first and second terms, but it becomes $-\frac{5}{12}$ between the third and fourth terms. This change in the common difference indicates that the sequence is not arithmetic.

What Type of Sequence is the Given Sequence?

Although the given sequence is not arithmetic, it is still a sequence of numbers. However, without further information about the sequence, it is difficult to determine its type. The sequence may be geometric, harmonic, or another type of sequence.

Conclusion

In conclusion, the given sequence $\frac{7}{12}, \frac{1}{6},-\frac{1}{4},-\frac{2}{3}$ is not arithmetic because the common difference is not constant. The common difference changes between the consecutive terms, indicating that the sequence is not arithmetic.

Answer to the Question

Based on the analysis above, the correct answer to the question is:

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

However, this answer is incorrect because the common difference is not constant throughout the sequence. The correct answer is that the sequence is not arithmetic.

Additional Information

For those who are interested in learning more about sequences and series, here are some additional resources:

Arithmetic Sequences: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Geometric Sequences: A geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is constant.
Harmonic Sequences: A harmonic sequence is a sequence of numbers in which the reciprocals of the terms form an arithmetic sequence.
Sequences and Series: Sequences and series are mathematical concepts that are used to describe the behavior of a sequence of numbers over time.

References

Mathematics Handbook: A comprehensive guide to mathematics, including sequences and series.
Calculus: A branch of mathematics that deals with the study of sequences and series.
Algebra: A branch of mathematics that deals with the study of equations and inequalities.

Final Thoughts

In the previous article, we analyzed the sequence $\frac{7}{12}, \frac{1}{6},-\frac{1}{4},-\frac{2}{3}$ and determined that it is not arithmetic. In this article, we will answer some frequently asked questions about sequences and series.

Q: What is the difference between a sequence and a series?

A: A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence. For example, the sequence $2, 4, 6, 8, 10$ is a list of numbers, while the series $2 + 4 + 6 + 8 + 10$ is the sum of the terms of the sequence.

Q: What is the formula for the nth term of a sequence?

A: The formula for the nth term of a sequence is given by:

$a_n = a_1 + (n-1)d$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is given by:

$S_n = \frac{a(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is given by:

$S_n = \frac{n}{2}(a_1 + a_n)$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Q: What is the formula for the sum of a harmonic series?

A: The formula for the sum of a harmonic series is given by:

$S_n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$

However, this series does not have a simple formula for its sum.

Q: What is the difference between a convergent and a divergent series?

A: A convergent series is a series whose sum approaches a finite value as the number of terms increases. A divergent series is a series whose sum approaches infinity as the number of terms increases.

Q: What is the formula for the sum of an infinite geometric series?

A: The formula for the sum of an infinite geometric series is given by:

$S = \frac{a}{1-r}$

where $S$ is the sum of the series, $a$ is the first term, and $r$ is the common ratio.

Q: What is the formula for the sum of an infinite arithmetic series?

A: The formula for the sum of an infinite arithmetic series is given by:

$S = \frac{a_1}{1-r}$

where $S$ is the sum of the series, $a_1$ is the first term, and $r$ is the common difference.

Q: What is the formula for the sum of an infinite harmonic series?

A: The formula for the sum of an infinite harmonic series is given by:

$S = \infty$

However, this series does not have a finite sum.

Conclusion

In conclusion, sequences and series are mathematical concepts that are used to describe the behavior of a sequence of numbers over time. The formulas for the sum of a geometric series, an arithmetic series, and a harmonic series are given above. The difference between a convergent and a divergent series is also explained above.

References

Mathematics Handbook: A comprehensive guide to mathematics, including sequences and series.
Calculus: A branch of mathematics that deals with the study of sequences and series.
Algebra: A branch of mathematics that deals with the study of equations and inequalities.

Final Thoughts

In conclusion, sequences and series are important mathematical concepts that are used to describe the behavior of a sequence of numbers over time. The formulas for the sum of a geometric series, an arithmetic series, and a harmonic series are given above. The difference between a convergent and a divergent series is also explained above.