The Need for a Recursive Rule

Oct 13, 2025 by wbrum.org 93 views

Understanding the Explicit Rule

The explicit rule for a sequence is given as: $a_n = 48 - 11n$ . This rule defines each term in the sequence as a function of its position, $n$ . The explicit rule is a clear and concise way to describe the sequence, but it may not be the most intuitive or useful way to work with the sequence, especially when it comes to finding previous terms or understanding the pattern of the sequence.

The Need for a Recursive Rule

A recursive rule, on the other hand, defines each term in the sequence as a function of the previous term(s). This type of rule is often more intuitive and easier to work with, especially when it comes to finding previous terms or understanding the pattern of the sequence. In this case, we are given the explicit rule and asked to find the recursive rule.

Finding the Recursive Rule

To find the recursive rule, we need to express each term in the sequence as a function of the previous term. Let's start by examining the explicit rule: $a_n = 48 - 11n$ . We can see that each term is defined in terms of the position, $n$ , but we need to find a way to express each term in terms of the previous term.

Option A: $a_n = a_{n-1} - 11$

One possible recursive rule is: $a_n = a_{n-1} - 11$ . This rule suggests that each term is equal to the previous term minus 11. To test this rule, let's see if it is consistent with the explicit rule. We can start by substituting $n-1$ for $n$ in the explicit rule:

$a_{n-1} = 48 - 11(n-1)$

Expanding the right-hand side, we get:

$a_{n-1} = 48 - 11n + 11$

Simplifying, we get:

$a_{n-1} = 59 - 11n$

Now, let's substitute this expression for $a_{n-1}$ into the recursive rule:

$a_n = a_{n-1} - 11$

$a_n = (59 - 11n) - 11$

Simplifying, we get:

$a_n = 48 - 11n$

This is consistent with the explicit rule, so option A is a possible recursive rule.

Option B: $a_n = a_{n-1} + 37$

Another possible recursive rule is: $a_n = a_{n-1} + 37$ . This rule suggests that each term is equal to the previous term plus 37. To test this rule, let's see if it is consistent with the explicit rule. We can start by substituting $n-1$ for $n$ in the explicit rule:

$a_{n-1} = 48 - 11(n-1)$

Expanding the right-hand side, we get:

$a_{n-1} = 48 - 11n + 11$

Simplifying, we get:

$a_{n-1} = 59 - 11n$

Now, let's substitute this expression for $a_{n-1}$ into the recursive rule:

$a_n = a_{n-1} + 37$

$a_n = (59 - 11n) + 37$

Simplifying, we get:

$a_n = 96 - 11n$

This is not consistent with the explicit rule, so option B is not a possible recursive rule.

Option C: $a_n = a_{n-1} - 37$

Another possible recursive rule is: $a_n = a_{n-1} - 37$ . This rule suggests that each term is equal to the previous term minus 37. To test this rule, let's see if it is consistent with the explicit rule. We can start by substituting $n-1$ for $n$ in the explicit rule:

$a_{n-1} = 48 - 11(n-1)$

Expanding the right-hand side, we get:

$a_{n-1} = 48 - 11n + 11$

Simplifying, we get:

$a_{n-1} = 59 - 11n$

Now, let's substitute this expression for $a_{n-1}$ into the recursive rule:

$a_n = a_{n-1} - 37$

$a_n = (59 - 11n) - 37$

Simplifying, we get:

$a_n = 22 - 11n$

This is not consistent with the explicit rule, so option C is not a possible recursive rule.

Conclusion

Based on our analysis, we have found that option A: $a_n = a_{n-1} - 11$ is a possible recursive rule for the sequence. This rule is consistent with the explicit rule and provides a clear and concise way to define each term in the sequence as a function of the previous term.

Recursive Rule for the Sequence

The recursive rule for the sequence is: $a_n = a_{n-1} - 11$ . This rule defines each term in the sequence as a function of the previous term and provides a clear and concise way to work with the sequence.

Initial Condition

To complete the recursive rule, we need to specify the initial condition, $a_1$ . We can find the initial condition by substituting $n=1$ into the explicit rule:

$a_1 = 48 - 11(1)$

Simplifying, we get:

$a_1 = 37$

Recursive Rule with Initial Condition

The recursive rule for the sequence with the initial condition is: $a_n = a_{n-1} - 11$ , $a_1 = 37$ . This rule defines each term in the sequence as a function of the previous term and provides a clear and concise way to work with the sequence.

Example

Let's use the recursive rule to find the first few terms of the sequence:

$a_1 = 37$

$a_2 = a_1 - 11 = 37 - 11 = 26$

$a_3 = a_2 - 11 = 26 - 11 = 15$

$a_4 = a_3 - 11 = 15 - 11 = 4$

Conclusion

In this article, we have found the recursive rule for the sequence: $a_n = a_{n-1} - 11$ , $a_1 = 37$ . This rule defines each term in the sequence as a function of the previous term and provides a clear and concise way to work with the sequence. We have also used the recursive rule to find the first few terms of the sequence.

Understanding the Recursive Rule

In our previous article, we found the recursive rule for the sequence: $a_n = a_{n-1} - 11$ , $a_1 = 37$ . This rule defines each term in the sequence as a function of the previous term and provides a clear and concise way to work with the sequence. In this article, we will answer some common questions about the recursive rule and the sequence.

Q: What is the recursive rule for the sequence?

A: The recursive rule for the sequence is: $a_n = a_{n-1} - 11$ , $a_1 = 37$ . This rule defines each term in the sequence as a function of the previous term.

Q: How do I use the recursive rule to find the next term in the sequence?

A: To use the recursive rule to find the next term in the sequence, simply substitute the previous term into the rule. For example, if we want to find the next term in the sequence after $a_1 = 37$ , we would substitute $a_1$ into the rule:

$a_2 = a_1 - 11$ $a_2 = 37 - 11$ $a_2 = 26$

Q: Can I use the recursive rule to find any term in the sequence?

A: Yes, you can use the recursive rule to find any term in the sequence. Simply substitute the previous term into the rule, and continue substituting until you reach the term you want to find.

Q: How do I know which term to start with when using the recursive rule?

A: When using the recursive rule, you need to start with the initial condition, which is $a_1 = 37$ in this case. This is the first term in the sequence, and it is used as the starting point for finding all subsequent terms.

Q: Can I use the recursive rule to find the value of a term that is not a whole number?

A: Yes, you can use the recursive rule to find the value of a term that is not a whole number. The recursive rule is a mathematical formula that can be used to find the value of any term in the sequence, regardless of whether it is a whole number or not.

Q: How do I know if the recursive rule is correct?

A: To check if the recursive rule is correct, you can use it to find the value of a term in the sequence and compare it to the value of that term found using the explicit rule. If the values are the same, then the recursive rule is correct.

Q: Can I use the recursive rule to find the value of a term that is not in the sequence?

A: No, you cannot use the recursive rule to find the value of a term that is not in the sequence. The recursive rule is a mathematical formula that is specific to the sequence, and it can only be used to find the value of terms that are part of the sequence.

Q: How do I use the recursive rule to find the value of a term that is a negative number?

A: To use the recursive rule to find the value of a term that is a negative number, simply substitute the previous term into the rule, just as you would for a positive number. The recursive rule will still work correctly, even if the term you are trying to find is a negative number.

Conclusion

In this article, we have answered some common questions about the recursive rule and the sequence. We have shown how to use the recursive rule to find the next term in the sequence, how to use it to find any term in the sequence, and how to check if the recursive rule is correct. We have also discussed some special cases, such as finding the value of a term that is not a whole number or finding the value of a term that is a negative number.