Join the IDNLearn.com community and start finding the answers you need today. Discover the reliable solutions you need with help from our comprehensive and accurate Q&A platform.
Sagot :
## Part a:
Let's analyze the given sequence first and find the rule:
### Sequence
[tex]\[ \begin{array}{|l|l|l|l|l|c|l|c|} \hline \text{Term} & 1 & 2 & 3 & 4 & \cdots & 17 & n \\ \hline \text{Value} & 1 & 6 & 36 & 216 & \cdots & ? & 5-\? \\ \hline \end{array} \][/tex]
### Values
1, 6, 36, 216
### Finding the Pattern
1. Term 1: [tex]\(1\)[/tex]
2. Term 2: [tex]\(6\)[/tex]
3. Term 3: [tex]\(36\)[/tex]
4. Term 4: [tex]\(216\)[/tex]
Notice the values increase rapidly. Let's express them in terms of the positions:
- [tex]\(1 = 1! \)[/tex]
- [tex]\(6 = 2! \)[/tex] ( factorial of 2 is 6)
- [tex]\(36 = 6 \times 6 = 6! / 5! \)[/tex]
- [tex]\(216 = 6 \times 6 \times 6 = 6^3\)[/tex]
Based on positions (n):
- [tex]\(1 = 1^1\)[/tex]
- [tex]\(6 = 2^2\)[/tex]
- [tex]\(36 = 3^3\)[/tex]
- [tex]\(216 = 4^4\)[/tex]
This indicates that each term's value [tex]\( a_n \)[/tex] can be described as [tex]\( n^n \)[/tex].
Rule:
[tex]\[ \text{Value}(n) = n^n \][/tex]
### Finding the 17th Term:
[tex]\[ \text{Value}(17) = 17^{17} \][/tex]
Since [tex]\( 17! \)[/tex] is extremely large, we can express it simply as [tex]\( 17^{17} \)[/tex].
### General Form for n-th Term:
[tex]\[ \text{Value}(n) = n^n \][/tex]
## Part b:
### Sequence
[tex]\[ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text{Term} & 1 & 2 & 3 & 4 & \cdots & 32 & n \\ \hline \text{Value} & 1 & 7 & 49 & 343 & \cdots & ? & ? \\ \hline \end{array} \][/tex]
### Values
1, 7, 49, 343
### Finding the Pattern
1. Term 1: [tex]\(1\)[/tex]
2. Term 2: [tex]\(7\)[/tex]
3. Term 3: [tex]\(49\)[/tex]
4. Term 4: [tex]\(343\)[/tex]
The differences between these terms give us:
- [tex]\(7 - 1 = 6\)[/tex]
- [tex]\(49 - 7 = 42\)[/tex]
- [tex]\(343 - 49 = 294\)[/tex]
Checking each by division:
- [tex]\(7 = 7^1\)[/tex]
- [tex]\(49 = 7^2\)[/tex]
- [tex]\(343 = 7^3\)[/tex]
We recognize the pattern. Let's express them in terms of the positions:
- [tex]\(1 = 7^0 \)[/tex]
- [tex]\(7 = 7^1 \)[/tex]
- [tex]\(49 = 7^2 \)[/tex]
- [tex]\(343 = 7^3 \)[/tex]
Rule:
[tex]\[ \text{Value}(n) = 7^{n-1} \][/tex]
### Finding the 32nd Term:
[tex]\[ \text{Value}(32) = 7^{31} \][/tex]
### General Form for n-th Term:
[tex]\[ \text{Value}(n) = 7^{n-1} \][/tex]
So for both sequences, we have:
### Sequence a:
- Rule: [tex]\( \text{Value}(n) = n^n \)[/tex]
- Example: 17th Term = 17^17
### Sequence b:
- Rule: [tex]\( \text{Value}(n) = 7^{n-1} \)[/tex]
- Example: 32nd Term = 7^{31}
Let's analyze the given sequence first and find the rule:
### Sequence
[tex]\[ \begin{array}{|l|l|l|l|l|c|l|c|} \hline \text{Term} & 1 & 2 & 3 & 4 & \cdots & 17 & n \\ \hline \text{Value} & 1 & 6 & 36 & 216 & \cdots & ? & 5-\? \\ \hline \end{array} \][/tex]
### Values
1, 6, 36, 216
### Finding the Pattern
1. Term 1: [tex]\(1\)[/tex]
2. Term 2: [tex]\(6\)[/tex]
3. Term 3: [tex]\(36\)[/tex]
4. Term 4: [tex]\(216\)[/tex]
Notice the values increase rapidly. Let's express them in terms of the positions:
- [tex]\(1 = 1! \)[/tex]
- [tex]\(6 = 2! \)[/tex] ( factorial of 2 is 6)
- [tex]\(36 = 6 \times 6 = 6! / 5! \)[/tex]
- [tex]\(216 = 6 \times 6 \times 6 = 6^3\)[/tex]
Based on positions (n):
- [tex]\(1 = 1^1\)[/tex]
- [tex]\(6 = 2^2\)[/tex]
- [tex]\(36 = 3^3\)[/tex]
- [tex]\(216 = 4^4\)[/tex]
This indicates that each term's value [tex]\( a_n \)[/tex] can be described as [tex]\( n^n \)[/tex].
Rule:
[tex]\[ \text{Value}(n) = n^n \][/tex]
### Finding the 17th Term:
[tex]\[ \text{Value}(17) = 17^{17} \][/tex]
Since [tex]\( 17! \)[/tex] is extremely large, we can express it simply as [tex]\( 17^{17} \)[/tex].
### General Form for n-th Term:
[tex]\[ \text{Value}(n) = n^n \][/tex]
## Part b:
### Sequence
[tex]\[ \begin{array}{|l|c|c|c|c|c|c|c|} \hline \text{Term} & 1 & 2 & 3 & 4 & \cdots & 32 & n \\ \hline \text{Value} & 1 & 7 & 49 & 343 & \cdots & ? & ? \\ \hline \end{array} \][/tex]
### Values
1, 7, 49, 343
### Finding the Pattern
1. Term 1: [tex]\(1\)[/tex]
2. Term 2: [tex]\(7\)[/tex]
3. Term 3: [tex]\(49\)[/tex]
4. Term 4: [tex]\(343\)[/tex]
The differences between these terms give us:
- [tex]\(7 - 1 = 6\)[/tex]
- [tex]\(49 - 7 = 42\)[/tex]
- [tex]\(343 - 49 = 294\)[/tex]
Checking each by division:
- [tex]\(7 = 7^1\)[/tex]
- [tex]\(49 = 7^2\)[/tex]
- [tex]\(343 = 7^3\)[/tex]
We recognize the pattern. Let's express them in terms of the positions:
- [tex]\(1 = 7^0 \)[/tex]
- [tex]\(7 = 7^1 \)[/tex]
- [tex]\(49 = 7^2 \)[/tex]
- [tex]\(343 = 7^3 \)[/tex]
Rule:
[tex]\[ \text{Value}(n) = 7^{n-1} \][/tex]
### Finding the 32nd Term:
[tex]\[ \text{Value}(32) = 7^{31} \][/tex]
### General Form for n-th Term:
[tex]\[ \text{Value}(n) = 7^{n-1} \][/tex]
So for both sequences, we have:
### Sequence a:
- Rule: [tex]\( \text{Value}(n) = n^n \)[/tex]
- Example: 17th Term = 17^17
### Sequence b:
- Rule: [tex]\( \text{Value}(n) = 7^{n-1} \)[/tex]
- Example: 32nd Term = 7^{31}
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
