Answer:
The first terms of this sequence should be:
f(1) = 65536
f(2) = 256
f(3) = 16
f(4) = 4
f(5) = 2
Explanation:
It is required to show the first tems of the sequence defined by the rule:
[tex]f(n) = \sqrt{f(n-1)}[/tex]
In this sense, it is required to calculate f(n-1) in order to calculate f(n).
First term f(1) = 65536 is given in the problem. So, we can calcualte the other terms by replacing in the recursive formulation for this sequence as follows:
[tex]f(1) =65536\\f(2) = \sqrt{f(1)}=\sqrt{65536} = 256\\f(2) = 256\\f(3) = \sqrt{f(2)}=\sqrt{256} =16\\f(3) = 16\\f(4) = \sqrt{f(3)}=\sqrt{16} =4\\f(4) = 4\\[/tex]
So we have that f(4) = 4.
Finally the term 5. f(5)
[tex]f(5)=\sqrt{f(4)}=\sqrt{4} \\f(5) = 2[/tex]
Finally, we can conclude that The first terms of this sequence should be:
f(1) = 65536
f(2) = 256
f(3) = 16
f(4) = 4
f(5) = 2
The first four terms of the sequence defined by the function [tex]f(n) = \sqrt{f(n - 1)} [/tex] are 65536, 256, 16 and 4 respectively
The terms in the sequence follows the rule defined by the function [tex]f(n) = \sqrt{f(n - 1)} [/tex]. Hence, the first four terms of the sequence for n = 1, 2, 3 and 4 can be calculated thus :
For n = 1 :
[tex]f(1) = 65536 [/tex]
For n = 2 :
[tex]f(2) = \sqrt{f(2 - 1)} = \sqrt{f(1)} = \sqrt{65536} = 256 [/tex]
For n = 3 :
[tex]f(3) = \sqrt{f(3 - 1)} = \sqrt{f(2)} = \sqrt{256} = 16 [/tex]
For n = 4 :
[tex]f(4) = \sqrt{f(4 - 1)} = \sqrt{f(3)} = \sqrt{16} = 4 [/tex]
Therefore, the first four terms of the sequence are 65536, 256, 16 and 4 respectively.
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