Answer:
If the sequence is arithmetic:
[tex]\boxed{\sf a_n=168n-328}[/tex]
[tex]\sf a_9=1184[/tex]
If the sequence is geometric:
[tex]\boxed{ \sf a_n=2(4)^{n-1}}[/tex]
[tex]\sf a_9=131072[/tex]
Step-by-step explanation:
An explicit formula allows you to find the nth term of a sequence.
As the question has not specified if the given terms are from an arithmetic or geometric sequence, I will provide answers for both.
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Explicit formula: Arithmetic sequence
[tex]\sf a_n=a+(n-1)d[/tex]
where:
- [tex]\sf a_n[/tex] is the nth term.
- a is the first term.
- n is the number of the term.
- d is the common difference.
Given:
- [tex]\sf a_2=8[/tex]
- [tex]\sf a_5=512[/tex]
Substitute the values into the formula to create two equations:
[tex]\begin{aligned}\sf a_2 & =\sf 8\\\sf a+(2-1)d&=\sf 8\\\sf a+d&=\sf 8\end{aligned}[/tex]
[tex]\begin{aligned}\sf a_5 & =\sf 512\\\sf a+(5-1)d&=\sf 512\\\sf a+4d&=\sf 512\end{aligned}[/tex]
Subtract the first equation from the second to eliminate a, and solve for d:
[tex]\begin{aligned}\implies \sf (a+4d)-(a+d) & = \sf 512-8\\\sf 3d & = \sf 504\\\sf d & = \sf 168\end{aligned}[/tex]
Substitute the found value of d into one of the equations and solve for a:
[tex]\begin{aligned}\sf a+d & = \sf 8\\\implies \sf a+168 & = \sf 8\\\sf a & = \sf -160\end{aligned}[/tex]
Substitute the found values of a and d into the formula to create an explicit formula:
[tex]\implies \sf a_n=-160+(n-1)168[/tex]
[tex]\implies \sf a_n=-160+168n-168[/tex]
[tex]\implies \sf a_n=168n-328[/tex]
To find a₉, substitute n = 9 into the found explicit formula:
[tex]\implies \sf a_9=168(9)-328[/tex]
[tex]\implies \sf a_9=1512-328[/tex]
[tex]\implies \sf a_9=1184[/tex]
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Explicit formula: Geometric sequence
[tex]\sf a_n=ar^{n-1}[/tex]
where:
- [tex]\sf a_n[/tex] is the nth term.
- a is the first term.
- n is the number of the term.
- r is the common ratio.
Given:
- [tex]\sf a_2=8[/tex]
- [tex]\sf a_5=512[/tex]
Substitute the values into the formula to create two equations:
[tex]\begin{aligned}\sf a_2 & =\sf 8\\\sf ar^{2-1}&=\sf 8\\\sf ar&=\sf 8\end{aligned}[/tex]
[tex]\begin{aligned}\sf a_5 & =\sf 512\\\sf ar^{5-1}&=\sf 512\\\sf ar^4&=\sf 512\end{aligned}[/tex]
Divide the second equation by the first to eliminate a, and solve for r:
[tex]\begin{aligned}\implies \sf \dfrac{ar^4}{ar} & = \sf \dfrac{512}{8}\\\sf r^{4-1} & = \sf 64\\ \sf r^3 & = \sf 64\\ \sf r & = \sf \sqrt[\sf 3]{\sf 64} \\ \sf r & = \sf 4\end{aligned}[/tex]
Substitute the found value of r into one of the equations and solve for a:
[tex]\begin{aligned}\sf ar & = \sf 8\\\implies \sf 4a & = \sf 8\\\sf a & = \sf 2\end{aligned}[/tex]
Substitute the found values of a and r into the formula to create an explicit formula:
[tex]\implies \sf a_n=2(4)^{n-1}[/tex]
To find a₉, substitute n = 9 into the found explicit formula:
[tex]\implies \sf a_9=2(4)^{9-1}[/tex]
[tex]\implies \sf a_9=2(4)^8[/tex]
[tex]\implies \sf a_9=131072[/tex]
Learn more about explicit formulas here:
https://brainly.com/question/27924553
