Question

Complete the following recursively defined functions.

Base case ?(0)=3

Recursive case ?(?) = 3?(? − 1) + 7 for n ≥ 1.

?(1) = ______

f(2) = _______

f(3) = ______

f(4) = ______

Base case ?(0)=1, ?(1)=2

Recursive case ?(?) = ?(? − 1)?(? − 2) for n ≥ 2.

g(2) = ______

g(3) = ______

g(4) = ______

g(5) = ______

Answer 1

So, as we have to give the value of the functions given in the question.

So, let's get started:

It is given that for n = 0, f(0) = 3,and the recursive function is, 

f(n) = 3f(n-1) + 7 for n>=1

Now, to find values of this function at different values we are going to put the value of n in the function:

Hence, f(1) = 3*f(1-1) + 7
            = 3*f(0) + 7 
            = 3*3 + 7               ----->(f(0) = 3, given)
            = 16.
Now, f(2) = 3*f(1) + 7
          = 3*16 + 7                ------>(f(1) = 16, we calculated)
          = 55
and, f(3) = 3*f(2) + 7
          = 3*55 + 7
          = 172
and f(4) = 3*f(3) + 7
         = 3*172 + 7
         =  523

         

So, this was all about the values of f for different values.

Now, let's discuss about the function g:

It is given that for n = 0, g(n) = 1 and for n = 1, g(n) = 2 ,and the recursive function is, 
g(n) = g(n-1)*g(n-2) for n>=2

Now, to find values of this function at different values we are going to put the value of n in the function:

Hence, g(2) = g(2-1)*g(2-2)
            = g(1)*g(0) 
            = 2 * 1                 ----->(g(0) = 1 and g(1) = 2, given)
            = 2.
Now, g(3) = g(2) * g(1)
          = 2 * 2                   ------>(g(1) = 2,given and g(2) = 2, we calculated)
          = 4
and, g(4) = g(3) * g(2)
          = 4 * 2
          = 8
and g(5) = g(4) * g(3) 
         = 8 * 4
         =  32

         

So, this was all about the values of g for different values.

So, this was the solutions of the problem.

I hope I am able to solve your problem, if yes then do give it a like.

It really helps :)