I didn't get the full answer, so I couldn't give you the answer you wanted. Sorry for that. And the question didn't say anyting about n is greater than 1. How do I make the equation?
Not getting a full answer (yet) is no reason not to do as we ask. We want to see your initial work, so we can encourage you and make suggestions for the next step. Math doesn't involve leaping directly to an answer; it is usually a matter of taking small steps.
Can't you at least follow the suggestion in #4 and try it out with some small numbers, then show us that work? At the very least, that will show us whether you understand the meaning of the problem. You might even discover the recurrence as you do it.
Here's a start:
Ahmet borrows n TL from Burak. Starting from today, Ahmet pays back to Burak 2 TL or 3 TL at each day. Let an be the number of ways Ahmet pays all of his debt to Burak. Find a recurrence relation for an. What are the initial conditions? What is a12?
Given a number n, we are to find how many sums (counting different orders as different) with terms 2 and 3 are possible.
For n=1, there are no ways.
For n=2, there is one way: 2.
For n=3, there is one way: 3.
For n=4, there is one way: 2+2.
For n=5, there are two ways: 2+3, 3+2.
Keep going,
paying attention to how you are thinking, specifically how you can use previous counts to find the next one. (For n=5, I could start either with 2 or 3, and then use existing answers to complete the sum to 5.) That's what a recurrence is.
Now,
think. (That is the entire point of a problem like this -- not to search the internet for an answer, or ask someone else.)
