Which function is correctly described by the recursive notation g(1)=3
, g(n−1)+3
for n>1
?(1 point)
Responses
g(n)=3n+1
g of n is equal to 3 n plus 1
g(n)=2n−1
g of n is equal to 2 n minus 1
g(n)=3n
g of n is equal to 3 n
g(n)=n+2
g(n)=2n−1
The correct function described by the recursive notation g(1)=3, g(n−1)+3 for n>1 is:
g(n) = 3n
To determine which function is correctly described by the given recursive notation g(1) = 3, g(n-1) + 3 for n > 1, we can follow these steps:
1. Start with the given base case, g(1) = 3. This means that when n is equal to 1, the function value is 3.
2. Use the recursive step, which states that g(n-1) + 3. This means that for any value of n greater than 1, the function value is equal to the value of g(n-1) plus 3.
3. By following the recursive step and substituting the value of g(n-1) in terms of g(n), we can derive the expression for the function.
Let's apply the recursive definition step by step:
For n = 1, g(1) = 3 (given)
For n = 2, g(2) = g(2-1) + 3 = g(1) + 3 = 3 + 3 = 6
For n = 3, g(3) = g(3-1) + 3 = g(2) + 3 = 6 + 3 = 9
For n = 4, g(4) = g(4-1) + 3 = g(3) + 3 = 9 + 3 = 12
Based on the pattern, we can observe that g(n) = 3n is correctly described by the given recursive notation.
So the correct function is: g(n) = 3n