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# Stepping Through Iterative Fibonacci Function : Understanding how the iterative fibonacci function works for a particular example

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- So we can make sure we really understand what's happening in this little iterative fibonacci fuction
- I am going to step through it with a particular example
- So we are going to assume that this function is called with the argument 5.
- So we want the 5th term in the fibonacci sequance.
- Starting at where indexing thing starting at 0.
- So the 0th term will be the... really... sometimes you would imagine the 1st term to be.
- Will start with the 0th term.
- So we're assuming that fibonacci of 5 is called.
- So we want the 5th term. Starting at the 0th term, so we want 5th one.
- And so when we go into this program terms starts... is defined as this list right here.
- That has just 2 elements: 0 and 1.
- So what I've done here is: I'm going really to focus what happens in this loop
- so as we enter into this loop the first time that we entered to this loop
- terms is refering to a list that has just 0 and 1 in it.
- And we did that because by definition those are the first two terms of the fibonacci sequence
- and then also entering into in the loop we defined i as beeing equal to... i as beeing equal to 2
- and we do that because we've already defined the 0th term and the 1st term
- and so now we want to as we enter the loop... start... we want to add...
- we want to add the second term as we go into the loop
- and then we say while i is less than or equal to n.
- Well we already know that n is 5.
- So in the scope of this function while we are running it, the variable n is 5.
- And so clearly i is still less than or equal to 5 so we will run this code right over here
- and we are appending to terms so we are going to append something.
- So entering into this clause is what i in terms would look like.
- But looks like we are going to append something to terms so terms is going to look like...
- Terms is going to be 0 and 1 and there we are going to add something...
- We are going to add something right over here to terms
- and what is that thing we are going to add?
- We are going to add whatever this bussiness over here is.
- It seems complicated, but when you break it down it doesn't look too bad.
- In this situation what is i minus 1?
- Well i is 2 so in this situation i minus 1 is going to be 1 and i minus 2 is going to be 0
- as we go trough this interation.
- So terms... the first term in terms... the first terms is the 0th term.
- This is the first term.
- So the first term in terms is literally a 1.
- So this whole thing is 1 and the 0th term in terms.
- Remember, terms... the 0th term in terms.
- Acctually I should write this way.
- 0th term in the... the list is called terms.
- 0th term in terms. This is what term looks like right now and it's going to be 0.
- So this whole thing is going to be 0 as well.
- So it's 1 plus 0 which is 1 and that's what we are appending to terms.
- So we are going to append a 1 over here and then we say i is equal to i plus 1.
- Well i is now equal to 2 so you are going to add 2 plus 1 is 3 and that's going to be the new value.
- The new value for i.
- And then we go back. We loop back up to the beginning of the while loop
- and we say while i is less than or equal to n.
- Well now i is a little bit closer to n. It's 3 now, but it's still less than or equal to 5.
- So now evaluate this again.
- Once again entering into the loop.
- Terms now looks like this: 0, 1, 1.
- i looks like this, 3.
- It's really what the same values we had exiting the loop
- and now we evaluate right over here.
- We are going to add something to terms.
- So terms right now: 0, 1, 1 we are going to add something to it.
- What is that something?
- It's going to be the i minus 1 term of terms.
- So what's the i?
- I over here is 3. 3 minus 1 is 2.
- So this right over here is now going to be 2.
- So it's going to be the second indexed term in terms so this is the 0th, the 1st the 2nd.
- So this over here is going to evaluate to 1 plus i minus 2.
- Well i is now 3. 3 minus 2 is 1. So plus the first term in terms.
- So this is the 0th term in the 1st term so this is also going to be a 1 over here.
- So it's going to be 1 plus 1 or 2.
- So we are going to append a 2 to terms.
- So that's what this does over here and then we take whatever i was
- which is i in this iteration is 3. We are going to add 1 to it and redefine i to be that.
- So 3 plus 1 is 4 and that's the new i.
- Then we go again to the beginning of the loop while i is less than or equal to n.
- While i is now 4, n is always, still 5.
- 4 is less than or equal to 5 so we run this again
- and once again we need to figure out what terms of i minus 1 are.
- Let me write it over here. This is getting kinda messy.
- Maybe i should clear this up. Cuz i wanna you to be able to read it clearly.
- So let me clear this.
- So now in this loop.
- Let's just think about what... so now... Let me write it over here.
- Entering the loop, now i is equal to 4.
- I'm going to do that in the same orange colour.
- Entering into the loop i is equal to 4 and the terms...
- Terms has now been... It now has 4 elements in this: 0, 1, 1 and 2.
- And now 4 is still less than or equal to 5 so we do this
- and we have to figure out what terms of i minus 1 is.
- Well now i minus 1 is 3. Right?
- 4 minus 1 is 3 so terms of... well the 3rd elem...
- not terms of. I should say the 3rd element in terms is: 0, 1, 2, 3.
- So it's going to be this whole thing.
- It's going to be this 2 right over here.
- So we are going to take that 2. I know you can read that.
- We are going to take this 2...
- This 2 and we are going to add it to terms, the i minus 2th terms.
- I is 4. 4 minus 2 is 2. The 2nd term in terms is 0, 1, 2.
- It's a 1.
- It's this one right over here.
- So we are adding this 2 plus this 1 to now get 3 and we are taking...
- So this whole thing... when you added 3 and we are appending it to terms.
- So terms was: 0, 1, 1, 2, but now we are appending a 3 to the end of it
- and then we are saying i is equal to i plus 1.
- I was 4. 4 plus 1 is 5 so that is what i is equal to now.
- It's equal to 5.
- Then we go to the beginning of the loop.
- I is now going to be equal to 5. Terms.
- Let me write that in the same colour for consistency.
- Terms are now: 0, 1, 1, 2 and 3 and then we say while i is less than or equal to n.
- N is 5. Well 5 is still less than or equal to 5.
- It's equal to 5 so this still is true so we'll execute this clause
- and now we have to figure out...
- Let me clear this out again.
- Now we have to figure out what is terms of i... what is the i minus 1'th term in terms.
- So now i is 5. so it's 5 minus 1. So it's the 4th term in terms.
- So 0, 1, 2, 3, 4. So it is this term right over here. It is the 3.
- So this over here is 3 now and then we have... and then we have to think about...
- Let me do this in another colour.
- Then we have i minus 2. I is 5 now. 5 minus 2 is 3. the 3rd term in terms: 0, 1, 2, 3 is this over here.
- So this over here is 2.
- So you evaluate this. You'll notice, we are just adding the last 2 terms that we had so far.
- This is how we build our fibonacci sequance
- and so 3 plus 2 is 5 so we are going to append a 5 to the end of terms.
- So terms is going to be: 0, 1, 1, 2, 3 and then we are going to append...
- We are going to append a 5 to it and then we say i is equal to i plus 1.
- So i is equal to 5 plus 1 or 6. I is equal to 6 now.
- When we go to the beginning of the loop and it says
- while i is less than or equal to n.
- Well now i is 6 and n is... has been 5. 6 is not less than or equal to 5 so this is false.
- So we do not... We break out of the loop and we go to...
- I guess we stopped running it and then we go to return...
- return the nth term in terms. So remember. N was 5.
- So what's the 5th term. If we started 0.
- So this is the 0th term, 1st term, 2nd term, 3rd term, 4th term, 5th term.
- And we are done.
- And hopefully that gives you and understanding of why this works and also a little bit of the logic
- of how we wrote it.
- It's literally building up the fibonacci sequence so the way you would expect to.
- It started with the first 2 terms by definition and then each time we went to the loop
- and added another term it said hey, the new term is going to be the sum of the last term right now
- and the second to last term and add them together and that will be the new term
- and you keep doing that until you have essentially... until you have added that nth term.

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