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Question regarding recursive functions

If someone could please provide some guidance on this I would greatly appreciate it. I'm taking a C++ Data Structures and Algorithms class, we are currently covering recursion.

The first part of the assignment was to create two programs using recursive functions from exercises from the text-book, no problem there. The second part is involves these jpeg attachments; they are mathematical proofs.

Apparently, I'm supposed to prove the proof in Lesson5_15.jpeg with a program using recursion. Then, I'm supposed to implement Lesson5_19.jpeg using mutual recursion.

I have asked several friends and co-workers who are Computer Science grads about this; they are as puzzled as I am. In fact for Lesson5_15.jpeg, I'm told the only way to actually prove it is through writing it out on paper. This is an online course taken through a local college and I never actually meet with the actual class and instrcutor everything is done through the internet.

The other students in my class are just as confused and to make matters worse, when I asked the instructor to be more specific with he wants, he proceeded to explain the theory behind these proofs, but nothing in regards to what he wants with them or even remotely on how to achieve this. I have emailed him again asking for further clarification and haven't heard anything for a couple days; I doubt I'll hear anything until Monday.

Please note, I'm not asking anyone to complete my assignment for me, just some input on what the instructor might want and how I might achieve this.

Thanks in advance,

Joe