What Is The Common Difference In The Following Arithmetic Sequence?$\[ \frac{2}{3}, \frac{1}{6}, -\frac{1}{3}, -\frac{5}{6}, \ldots \\]A. \[$-4\$\]B. \[$-\frac{1}{2}\$\]C. \[$\frac{1}{9}\$\]D. \[$4\$\]

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Understanding Arithmetic Sequences

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 are an essential concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics.

Identifying the Common Difference

To find the common difference in an arithmetic sequence, we need to examine the given sequence and identify the pattern. In the given sequence, the terms are: 23,16,−13,−56,…\frac{2}{3}, \frac{1}{6}, -\frac{1}{3}, -\frac{5}{6}, \ldots. We can start by finding the difference between the first two terms:

16−23=−12\frac{1}{6} - \frac{2}{3} = -\frac{1}{2}

However, we need to find the common difference, which is the constant difference between any two consecutive terms. To do this, we can find the difference between the second and third terms:

−13−16=−12-\frac{1}{3} - \frac{1}{6} = -\frac{1}{2}

As we can see, the difference between the second and third terms is the same as the difference between the first and second terms. This confirms that the given sequence is an arithmetic sequence with a common difference of −12-\frac{1}{2}.

Verifying the Common Difference

To verify that the common difference is indeed −12-\frac{1}{2}, we can find the difference between the third and fourth terms:

−56−(−13)=−56+13=−46+26=−26=−13-\frac{5}{6} - (-\frac{1}{3}) = -\frac{5}{6} + \frac{1}{3} = -\frac{4}{6} + \frac{2}{6} = -\frac{2}{6} = -\frac{1}{3}

However, we are looking for the common difference, which is the difference between any two consecutive terms. To find the common difference, we can find the difference between the fourth and fifth terms:

Since the fifth term is not given, we cannot find the difference between the fourth and fifth terms. However, we can use the fact that the common difference is the same throughout the sequence to find the fifth term.

Finding the Fifth Term

To find the fifth term, we can add the common difference to the fourth term:

−56+(−12)=−56−36=−86=−43-\frac{5}{6} + (-\frac{1}{2}) = -\frac{5}{6} - \frac{3}{6} = -\frac{8}{6} = -\frac{4}{3}

Now that we have the fifth term, we can find the difference between the fourth and fifth terms:

−43−(−56)=−43+56=−86+56=−36=−12-\frac{4}{3} - (-\frac{5}{6}) = -\frac{4}{3} + \frac{5}{6} = -\frac{8}{6} + \frac{5}{6} = -\frac{3}{6} = -\frac{1}{2}

As we can see, the difference between the fourth and fifth terms is indeed −12-\frac{1}{2}, which confirms that the common difference is −12-\frac{1}{2}.

Conclusion

In conclusion, the common difference in the given arithmetic sequence is −12-\frac{1}{2}. This can be verified by finding the difference between any two consecutive terms, which is always −12-\frac{1}{2}.

Frequently Asked Questions

  • What is an arithmetic sequence? An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
  • What is the common difference in an arithmetic sequence? The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.
  • How do I find the common difference in an arithmetic sequence? To find the common difference, you can find the difference between any two consecutive terms in the sequence.

Final Answer

The final answer is −12\boxed{-\frac{1}{2}}.

Understanding Arithmetic Sequences

Arithmetic sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will answer some of the most frequently asked questions about arithmetic sequences.

Q&A

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. This constant difference is known as the common difference.

Q: What is the common difference in an arithmetic sequence?

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

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

A: To find the common difference, you can find the difference between any two consecutive terms in the sequence.

Q: What is the formula for finding the nth term of an arithmetic sequence?

A: The formula for finding the nth term of an arithmetic sequence is given by:

an=a1+(n−1)da_n = a_1 + (n-1)d

where ana_n is the nth term, a1a_1 is the first term, nn is the term number, and dd is the common difference.

Q: How do I find the sum of the first n terms of an arithmetic sequence?

A: To find the sum of the first n terms of an arithmetic sequence, you can use the formula:

Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)

where SnS_n is the sum of the first n terms, a1a_1 is the first term, ana_n is the nth term, and nn is the term number.

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

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

S=a11−rS = \frac{a_1}{1 - r}

where SS is the sum, a1a_1 is the first term, and rr is the common ratio.

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

A: To determine if a sequence is an arithmetic sequence, you can check if the difference between any two consecutive terms is constant. If it is, then the sequence is an arithmetic sequence.

Q: What are some real-world applications of arithmetic sequences?

A: Arithmetic sequences have many real-world applications, including:

  • Finance: Arithmetic sequences are used to calculate interest rates and investment returns.
  • Music: Arithmetic sequences are used to create musical scales and rhythms.
  • Science: Arithmetic sequences are used to model population growth and decay.

Conclusion

In conclusion, arithmetic sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics. By answering some of the most frequently asked questions about arithmetic sequences, we hope to have provided a better understanding of this important concept.

Final Answer

The final answer is 0\boxed{0}.

Additional Resources

Arithmetic Sequence Formula

The formula for finding the nth term of an arithmetic sequence is given by:

an=a1+(n−1)da_n = a_1 + (n-1)d

where ana_n is the nth term, a1a_1 is the first term, nn is the term number, and dd is the common difference.

Arithmetic Sequence Examples

  • Find the 5th term of the arithmetic sequence: 2, 5, 8, 11, ...
  • Find the sum of the first 10 terms of the arithmetic sequence: 1, 4, 7, 10, ...
  • Find the sum of the infinite arithmetic sequence: 1, 2, 3, 4, ...

Arithmetic Sequence Applications

  • Finance: Arithmetic sequences are used to calculate interest rates and investment returns.
  • Music: Arithmetic sequences are used to create musical scales and rhythms.
  • Science: Arithmetic sequences are used to model population growth and decay.