Respuesta :
Problem 1)
The notation you wrote down is a bit odd. There seems to be a typo. Please double check.
Assuming you meant to write
[tex]a_{n} = 3 a_{n-1}[/tex]
then the answer is choice D) 2, 6, 18, 54, 162, ...
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Why is this? Because each term is found by multiplying the previous term by 3
first term = 2
second term = (first term)*3 = 2*3 = 6
third term = (second term)*3 = 6*3 = 18
fourth term = (third term)*3 = 18*3 = 54
fifth term = (fourth term)*3 = 54*3 = 162
this pattern continues forever
Extra information: The closed form equation is [tex]a_{n} = 2(3)^{n-1}[/tex] where 2 is the first term, 3 is the common ratio, and n is a positive integer.
some more info: notice picking any term in sequence {2,6,18,54,162,...} and dividing it over its previous term leads to 3
6/2 = 3
18/6 = 3
54/18 = 3
162/54 = 3
None of the other sequences listed have this property of the adjacent terms dividing to a constant ratio. So none of the other sequences are geometric.
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Problem 2)
The first term is [tex]t_{1} = 2[/tex] (ie, the first term is 2)
The recursive definition is [tex]t_{n+1} = 3^{t_{n}}[/tex] basically telling us "to get the next term, raise the prior term as an exponent with 3 as the base"
The general template looking like this: nth term = 3^(term just before the nth term)
The first term is 2, so the next term must be 3^2 = 9
The third term would be 3^9 = 19683 and so on.
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Final Answer: choice B) 9
Answer
an=3+an-1
2, 5, 14, 41, 122..
9
Step-by-step explanation:
