The Explicit Rule For An Arithmetic Sequence Is A N = 20 3 + 1 3 ( N − 1 A_n = \frac{20}{3} + \frac{1}{3}(n-1 A N ​ = 3 20 ​ + 3 1 ​ ( N − 1 ]. What Is The Value Of The 89 Th 89^{\text{th}} 8 9 Th Term?A. 89 3 \frac{89}{3} 3 89 ​ B. 36 C. 287 3 \frac{287}{3} 3 287 ​ D. 248

Understanding the Explicit Rule for an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. The explicit rule for an arithmetic sequence is given by the formula $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. However, in this problem, we are given a different explicit rule, which is $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$. This rule provides a direct way to find the value of any term in the sequence without having to know the first term or the common difference.

Breaking Down the Explicit Rule

To understand how the explicit rule works, let's break it down into its components. The rule is given by $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$. The first term, $\frac{20}{3}$, is the initial value of the sequence, and the second term, $\frac{1}{3}(n-1)$, represents the common difference multiplied by the term number minus one. This means that each term in the sequence is obtained by adding the common difference to the previous term.

Finding the Value of the 89th Term

Now that we have a clear understanding of the explicit rule, we can use it to find the value of the 89th term. To do this, we simply need to substitute $n=89$ into the explicit rule and evaluate the expression.

Step 1: Substitute n=89 into the Explicit Rule

$a_{89} = \frac{20}{3} + \frac{1}{3}(89-1)$

Step 2: Simplify the Expression

$a_{89} = \frac{20}{3} + \frac{1}{3}(88)$

Step 3: Evaluate the Expression

$a_{89} = \frac{20}{3} + \frac{88}{3}$

$a_{89} = \frac{108}{3}$

$a_{89} = 36$

Conclusion

The value of the 89th term in the arithmetic sequence is 36. This can be verified by using the explicit rule for an arithmetic sequence and substituting $n=89$ into the formula.

Comparison with the Given Options

The value of the 89th term, 36, matches option B. This confirms that the correct answer is indeed 36.

Final Thoughts

In this problem, we used the explicit rule for an arithmetic sequence to find the value of the 89th term. The explicit rule provided a direct way to find the value of any term in the sequence without having to know the first term or the common difference. By breaking down the explicit rule and simplifying the expression, we were able to find the value of the 89th term, which is 36. This problem demonstrates the importance of understanding the explicit rule for an arithmetic sequence and how it can be used to find the value of any term in the sequence.

Understanding the Explicit Rule for an Arithmetic Sequence

In our previous article, we discussed the explicit rule for an arithmetic sequence and how it can be used to find the value of any term in the sequence. However, we understand that there may be some questions and concerns that readers may have. In this article, we will address some of the most frequently asked questions about the explicit rule for an arithmetic sequence.

Q&A

Q: What is the explicit rule for an arithmetic sequence?

A: The explicit rule for an arithmetic sequence is given by the formula $a_n = a_1 + (n-1)d$, where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference. However, in this problem, we are given a different explicit rule, which is $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$.

Q: How do I find the value of any term in the sequence using the explicit rule?

A: To find the value of any term in the sequence using the explicit rule, you simply need to substitute the term number into the formula and evaluate the expression. For example, to find the value of the 89th term, you would substitute $n=89$ into the formula and evaluate the expression.

Q: What is the difference between the explicit rule and the recursive rule for an arithmetic sequence?

A: The explicit rule and the recursive rule for an arithmetic sequence are two different ways of representing the sequence. The explicit rule provides a direct way to find the value of any term in the sequence, while the recursive rule provides a way to find the value of any term in the sequence by using the previous term.

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

A: The common difference in an arithmetic sequence is the difference between any two consecutive terms. To determine the common difference, you can use the formula $d = a_n - a_{n-1}$, where $d$ is the common difference, $a_n$ is the nth term, and $a_{n-1}$ is the previous term.

Q: Can I use the explicit rule to find the value of any term in a geometric sequence?

A: No, the explicit rule for an arithmetic sequence cannot be used to find the value of any term in a geometric sequence. The explicit rule for a geometric sequence is given by the formula $a_n = a_1 \cdot r^{n-1}$, where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Q: How do I use the explicit rule to find the value of the first term in an arithmetic sequence?

A: To use the explicit rule to find the value of the first term in an arithmetic sequence, you can substitute $n=1$ into the formula and evaluate the expression. For example, if the explicit rule is $a_n = \frac{20}{3} + \frac{1}{3}(n-1)$, then the value of the first term is $a_1 = \frac{20}{3} + \frac{1}{3}(1-1) = \frac{20}{3}$.

Conclusion

In this article, we addressed some of the most frequently asked questions about the explicit rule for an arithmetic sequence. We hope that this article has provided a better understanding of the explicit rule and how it can be used to find the value of any term in the sequence. If you have any further questions or concerns, please don't hesitate to contact us.

Final Thoughts

The explicit rule for an arithmetic sequence is a powerful tool that can be used to find the value of any term in the sequence. By understanding the explicit rule and how it can be used, you can solve a wide range of problems involving arithmetic sequences. We hope that this article has provided a useful resource for anyone looking to learn more about arithmetic sequences and the explicit rule.