## 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}
