Which function is correctly described by the recursive notation g (1) = 3,
g (n - 1) + 3 for n > 1? (1 point)
O g (n) = 31
O g (n) = 3n +1
O g (n) = 2n - 1
O g (n) = n + 2
The correct option is:
O g (n) = 3n +1
The recursive notation g(n) = g(n-1) + 3 for n > 1, with the base case g(1) = 3, can be used to define the function g(n) recursively.
To find a closed-form expression for g(n), we can expand the recursive definition:
g(n) = g(n-1) + 3
= (g(n-2) + 3) + 3
= g(n-2) + 2(3)
= (g(n-3) + 3) + 2(3)
= g(n-3) + 3(2)
= g(n-4) + 3 + 3(2)
= g(n-4) + 4(3)
= ...
We can observe that as we expand the recursive definition, the expression 3k (where k is the number of recursive steps) appears.
So, for the kth recursive step, we have g(n) = g(n-k) + k(3).
Using this observation, we can find a general expression for g(n).
When k = n-1 (the maximum value of k for this recursive definition), we have:
g(n) = g(n - (n - 1)) + (n - 1)(3)
= g(1) + (n - 1)(3)
= 3 + (n - 1)(3)
= 3 + 3(n - 1)
= 3 + 3n - 3
= 3n
Therefore, the function g(n) is correctly described by the recursive notation:
g(n) = 3n
So the correct option is: g(n) = 3n + 1 (Choice B).
To find the correct function for the given recursive notation, let's analyze each option based on the given information.
Option A: g(n) = 31
This option doesn't match the given recursive notation. The recursive relation given is g(n - 1) + 3, which indicates that g(n) is dependent on g(n - 1), not a constant value like 31. Therefore, Option A can be eliminated.
Option B: g(n) = 3n + 1
This option matches the given recursive notation. The recursive relation g(n - 1) + 3 implies that each subsequent term is obtained by adding 3 to the previous term. By observing the pattern, it can be deduced that g(n) = 3n + 1 is the correct function.
Option C: g(n) = 2n - 1
This option doesn't match the given recursive notation. The recursive relation given is g(n - 1) + 3, which involves adding 3 to the previous term, not multiplying it by 2. Therefore, Option C can be eliminated.
Option D: g(n) = n + 2
This option doesn't match the given recursive notation either. The recursive relation g(n - 1) + 3 involves adding 3 to the previous term but not simply incrementing by 1. Therefore, Option D can be eliminated.
Therefore, the correct function described by the recursive notation g(1) = 3, g(n - 1) + 3 for n > 1 is:
Option B: g(n) = 3n + 1.