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 substitute $n = 89$ into the explicit rule and evaluate the expression. This gives us:

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

Simplifying the Expression

To simplify the expression, we first evaluate the term inside the parentheses:

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

Next, we multiply the common difference by the term number minus one:

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

Combining the Terms

Now that we have the two terms, we can combine them by adding them together:

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

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

Evaluating the Expression

Finally, we can evaluate the expression by dividing the numerator by the denominator:

$a_{89} = 36$

Conclusion

In this problem, we used the explicit rule for an arithmetic sequence to find the value of the 89th term. By substituting $n = 89$ into the explicit rule and evaluating the expression, we found that the value of the 89th term is 36. This demonstrates the power of the explicit rule in finding the value of any term in an arithmetic sequence.

Comparison of Options

Let's compare our answer with the options provided:

A. $\frac{89}{3}$

B. 36

C. $\frac{287}{3}$

D. 248

Our answer, 36, matches option B. Therefore, the correct answer is B.

Final Thoughts

In conclusion, the explicit rule for an arithmetic sequence provides a direct way to find the value of any term in the sequence. By using the explicit rule, we can easily find the value of the 89th term, which is 36. This problem demonstrates the importance of understanding the explicit rule and how to use it to solve problems involving arithmetic sequences.

Understanding the Explicit Rule for an Arithmetic Sequence

In our previous article, we discussed the explicit rule for an arithmetic sequence and how to use it to find the value of any term in the sequence. In this article, we will answer some frequently asked questions about the explicit rule and provide additional examples to help solidify your understanding.

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.

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

A: 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 by using the previous term.

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

A: To use the explicit rule, simply substitute the term number and the common difference into the formula and evaluate the expression.

Q: What if I don't know the first term or the common difference?

A: If you don't know the first term or the common difference, you can use the explicit rule to find the value of any term by using the formula $a_n = a_1 + (n-1)d$ and solving for $a_1$ or $d$.

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

A: No, the explicit rule is only used for arithmetic sequences. For geometric sequences, you would use 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: What if I have a sequence with a non-constant common difference?

A: If you have a sequence with a non-constant common difference, you would need to use a different formula to find the value of any term in the sequence.

Examples

Example 1: Finding the Value of the 10th Term

Find the value of the 10th term in the sequence with the explicit rule $a_n = 2 + (n-1)3$.

Solution

To find the value of the 10th term, we substitute $n = 10$ into the explicit rule and evaluate the expression:

$a_{10} = 2 + (10-1)3$

$a_{10} = 2 + 9 \cdot 3$

$a_{10} = 2 + 27$

$a_{10} = 29$

Example 2: Finding the Value of the 5th Term

Find the value of the 5th term in the sequence with the explicit rule $a_n = 5 + (n-1)2$.

Solution

To find the value of the 5th term, we substitute $n = 5$ into the explicit rule and evaluate the expression:

$a_{5} = 5 + (5-1)2$

$a_{5} = 5 + 4 \cdot 2$

$a_{5} = 5 + 8$

$a_{5} = 13$

Conclusion

In this article, we answered some frequently asked questions about the explicit rule for an arithmetic sequence and provided additional examples to help solidify your understanding. We also discussed the differences between the explicit rule and the recursive rule, and how to use the explicit rule to find the value of any term in the sequence. By following the steps outlined in this article, you should be able to use the explicit rule to find the value of any term in an arithmetic sequence.

Final Thoughts

In conclusion, the explicit rule for an arithmetic sequence provides a direct way to find the value of any term in the sequence. By using the explicit rule, you can easily find the value of any term in an arithmetic sequence, making it a powerful tool for solving problems involving arithmetic sequences.