Understanding the Explicit Rule for an Arithmetic Sequence

Oct 13, 2025 by talynt.org 130 views

Understanding the Explicit Rule for an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members 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 nth term of 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 states that the nth term of the sequence is equal to $\frac{20}{3} + \frac{1}{3}(n-1)$ . The first part of the rule, $\frac{20}{3}$ , represents the first term of the sequence. The second part of the rule, $\frac{1}{3}(n-1)$ , represents the common difference multiplied by the term number minus one.

Finding the 89th Term

Now that we understand the explicit rule, we can use it to find the 89th term of the sequence. To do this, we simply substitute $n=89$ into the rule and evaluate the expression.

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

Simplifying the Expression

To simplify the expression, we first evaluate the expression inside the parentheses.

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

Multiplying the Common Difference by the Term Number

Next, we multiply the common difference by the term number.

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

Adding the First Term and the Common Difference

Finally, we add the first term and the common difference to get the 89th term.

$a_{89} = \frac{20}{3} + \frac{88}{3} = \frac{108}{3} = 36$

Conclusion

In conclusion, the value of the 89th term of the arithmetic sequence is 36. This is the correct answer, and it can be verified by plugging in the value of $n=89$ into the explicit rule and evaluating the expression.

Final Answer

The final answer is 36.

Understanding the Explicit Rule for an Arithmetic Sequence

In the previous article, we discussed the explicit rule for an arithmetic sequence and how to use it to find the nth term of 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: 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 use the explicit rule to find the nth term of the sequence?

A: To use the explicit rule to find the nth term of the sequence, simply substitute the value of $n$ into the rule and evaluate the expression. For example, to find the 89th term of the sequence, we would substitute $n=89$ into the rule and evaluate the expression.

Q: What is the first term of the sequence?

A: The first term of the sequence is $\frac{20}{3}$ .

Q: What is the common difference of the sequence?

A: The common difference of the sequence is $\frac{1}{3}$ .

Q: How do I find the common difference of the sequence?

A: To find the common difference of the sequence, we can use the explicit rule and substitute the value of $n$ into the rule. For example, to find the common difference of the sequence, we would substitute $n=2$ into the rule and evaluate the expression.

Q: What is the value of the 89th term of the sequence?

A: The value of the 89th term of the sequence is 36.

Q: How do I verify the value of the 89th term of the sequence?

A: To verify the value of the 89th term of the sequence, we can plug in the value of $n=89$ into the explicit rule 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 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. The recursive rule for an arithmetic sequence is given by the formula $a_n = a_{n-1} + d$ , where $a_n$ is the nth term and $d$ is the common difference.

Q: How do I use the recursive rule to find the nth term of the sequence?

A: To use the recursive rule to find the nth term of the sequence, we can start with the first term and add the common difference repeatedly until we reach the desired term number.

Q: What are some common applications of arithmetic sequences?

A: Arithmetic sequences have many common applications in mathematics, science, and engineering. Some examples include:

Modeling population growth
Modeling financial investments
Modeling physical phenomena such as the motion of objects
Creating musical scales and rhythms

Q: How do I use arithmetic sequences to model real-world phenomena?

A: To use arithmetic sequences to model real-world phenomena, we can start by identifying the pattern of the sequence and then use the explicit or recursive rule to find the nth term of the sequence.

Q: What are some common mistakes to avoid when working with arithmetic sequences?

A: Some common mistakes to avoid when working with arithmetic sequences include:

Confusing the explicit rule with the recursive rule
Forgetting to evaluate the expression when using the explicit rule
Not checking the value of the common difference
Not verifying the value of the nth term

Conclusion

In conclusion, the explicit rule for an arithmetic sequence is a powerful tool for finding the nth term of the sequence. By understanding the explicit rule and how to use it, we can model real-world phenomena and make predictions about future events. Remember to always verify the value of the nth term and to check the value of the common difference to avoid common mistakes.