Alright, let's work through each of these exercises step-by-step to find the indicated terms of the sequences.
### Exercise 5
Given sequence: [tex]\( a_n = \left(\frac{1}{2}\right)^n \)[/tex]
To find: [tex]\( a_9 \)[/tex]
Plug [tex]\( n = 9 \)[/tex] into the formula:
[tex]\[ a_9 = \left(\frac{1}{2}\right)^9 \][/tex]
[tex]\[ a_9 = \frac{1}{2^9} \][/tex]
[tex]\[ a_9 = \frac{1}{512} \][/tex]
Converting to decimal form:
[tex]\[ a_9 \approx 0.001953125 \][/tex]
So, [tex]\( a_9 = 0.001953125 \)[/tex].
### Exercise 6
Given sequence: [tex]\( a_n = \frac{(n+1)^2}{n-9} \)[/tex]
To find: [tex]\( a_{14} \)[/tex]
Plug [tex]\( n = 14 \)[/tex] into the formula:
[tex]\[ a_{14} = \frac{(14+1)^2}{14-9} \][/tex]
[tex]\[ a_{14} = \frac{15^2}{5} \][/tex]
[tex]\[ a_{14} = \frac{225}{5} \][/tex]
[tex]\[ a_{14} = 45 \][/tex]
So, [tex]\( a_{14} = 45.0 \)[/tex].
### Exercise 7
Given sequence: [tex]\( a_n = \frac{(-1)^{n+1}(n-1)(n+2)}{n} \)[/tex]
To find: [tex]\( a_7 \)[/tex]
Plug [tex]\( n = 7 \)[/tex] into the formula:
[tex]\[ a_7 = \frac{(-1)^{7+1}(7-1)(7+2)}{7} \][/tex]
[tex]\[ a_7 = \frac{(-1)^8 \cdot 6 \cdot 9}{7} \][/tex]
Since [tex]\((-1)^8 = 1\)[/tex],
[tex]\[ a_7 = \frac{1 \cdot 6 \cdot 9}{7} \][/tex]
[tex]\[ a_7 = \frac{54}{7} \][/tex]
[tex]\[ a_7 \approx 7.714285714285714 \][/tex]
So, [tex]\( a_7 \approx 7.714285714285714 \)[/tex].
### Exercise 8
Given sequence: [tex]\( a_n = \left(\frac{n}{9} - 12\right)^n \)[/tex]
To find: [tex]\( a_{99} \)[/tex]
Plug [tex]\( n = 99 \)[/tex] into the formula:
[tex]\[ a_{99} = \left(\frac{99}{9} - 12\right)^{99} \][/tex]
[tex]\[ a_{99} = \left(11 - 12\right)^{99} \][/tex]
[tex]\[ a_{99} = (-1)^{99} \][/tex]
Since [tex]\( (-1)^{99} = -1 \)[/tex],
[tex]\[ a_{99} = -1 \][/tex]
So, [tex]\( a_{99} = -1.0 \)[/tex].
In summary, the indicated terms for the sequences are:
1. [tex]\( a_9 = 0.001953125 \)[/tex]
2. [tex]\( a_{14} = 45.0 \)[/tex]
3. [tex]\( a_7 = 7.714285714285714 \)[/tex]
4. [tex]\( a_{99} = -1.0 \)[/tex]
