Now, in fact, there never was such an m, and hence there never was such an n.

Because the conclusion as a result is valid.

But within the text of this argument, these guys are specific.

Here, there's a, there's a, this is a variable that's quantified, alright?

In any case this is proof this is true. On number four, we have to use the course

rubric again. first of all, let's see what, what the

theorem says. It says that if we take the Fibonacci

numbers. And we square them and we take the first

n of them and square them another. Then the result is equal to the nth

Fibbionaci number, the last one in this sequence here, multiplied by the one

after that. now this is actually, this is true, by

the way. This is true, it's a valid theorem.

It's one of many identities about Fibonacci numbers that show that they're

connected in In what at first, a very surprising ways.

they're very typical in, that they're almost always proved by induction.

And when you look at the induction proof, you realize that it's really that the

identity, which at first, seems surprising.

Is actually just a disguised version of the definition of the Fibonacci numbers.

What's behind the, all of these interesting identities.

Is the fact that the the n plus second Fibonacci number is the sum of the nth

Fibonacci number and the n plus first Fibonacci number.

So they all go back to the fact that it's defined in this iteratively additive way,

okay? well, let's just see if this one, see,

see how this one goes, okay? first of all, we'll look at logical

correctness. worrying about things like reasons and so

forth later. Let's check that the, the first case is

true. F1 is equal to one, so F1 squared, and in

the case, n equals one, there is no sum it's just F1 squared.

So the left hand side is just one, on the right hand side is F1, which is one.

And of course the second Fibonacci number is also one.

Because the Fibonacci sequence begins with a pair of ones.

So we have two ones on the right. So, this is logically correct.

then there's the, the induction step. Let's just check the algebra here.

here it's just the taking taking a sum up to n plus one, and pulling out the second

one, the last one, the last term in the series.

So we've got the sum of n plus 1 is the sum of the first n together with the last

one. we'll look at the issue of, of reasons in

a minute. this will be the induction hypothesis,

and nicely stated as, as a reason. that this sum equals Fn plus one, this is

just Fn plus one squared I carried through.

take out Fn plus one as a common factor okay.

let's see, this is definition of Fn plus two, is the sum of that one plus that

one. And then we've, we've got the other

identity n plus 1. Okay, so, so all of the logical steps are

correct. This is a valid induction proof.

So I'm going to give four marks for that. is it clear?

Yes, I think this is clear. again, there's going to be some issues of

reason as to explain things, but everything is clear.

It's well laid out, it's easy to follow steps, even when I had to figure out what

the author meant. I'm going to, we'll talk in a minute as

to whether, whether I should have had to. I, it, it was, it was easy to follow in

that sense. So it, so it was clear.

there's an opening, it's well opened. It's a proof by induction, there's a

standard method. And, and they, they, the good way to

start. The correct, well, almost to start to

prove using a standard method is to state what the method is.

So I want to get full marks for that. conclusion when the conclusion is stated

I'll, I'll look at whether it's properly stated in a minute.

But it certainly it, it, stated when the proof is complete And it's been laid out

that it's going to be an induction proof. So, I think we're going to get four for

that reasons, a couple of quibbles. I think the order should have said,

separate out fine term. this is good, stating the, the use of the

induction hypothesis and an induction proof is, is always, is critical.

I, I, I think that's this is such an important step.

by algebra, yeah, you could say something like check out a common factor but the,

this is, this level of mathematics. When we're doing proofs in number theory

as I mentioned with the last problem set. we, we can assume that people can, can

spot things like taking out common factors.

this I think is important. and this, this this is critical, the fact

that we're using the, the definition of the, of the Fibonacci numbers.

the, these kinds of identities, as I mentioned a moment ago, these kind of

identities actually only hold because of the way the Fibonacci sequence's identity

is, is defined. which establishes the identity for n plus

one. That's good that's a local conclusion,

which is good. the proof is complete, here the person

should have said, by induction, or by the principle of induction.

let's put out in four principle of induction, because there is a powerful

fact about what the, the natural numbers that's been used here.

Okay, so, what I'm going to do for reasons, I'm going to give two, I think.

Because I can't give four. you know, I, if, if this one arguably I,

I, I would tend to lean on having this in especially since it's an opening step in

the proof. But if that was the only thing that was

missing in this context, I might not have even deducted any marks at all.

this one however, I, I have to deduct at least one, and I think, I think I've

really deducted, I think really I've deducted the two for this one.

It was, it was a bit of a judgement call. you know, I have to allow for the fact

that the author of a proof maybe made a slightly different judgement call.

you, you have to try, judge how well you think a persons putting down a proof.

and, and you can't really say, I always do it this way, therefore you always

should do it this way. Because people have different, they come

on different sides of these issues. we're really looking to grade this as an

overall thing. You know, one of the problems with using

a rubric is we're trying to take something that's holistic.

And is basically an overall judgement call and reduce it to number of

parameters. This is not how professionals go ,they

look overall and say ,this is a good proof and then assign a number ,but now

space for years of expertise. Pulling us apart this way is a good for

beginners because it allows you to focus on one only individual things, but a

professional looks at all these things in, in one.

And you have to sort of balance things out.

and that's why we're going to all these videos, to try and give you some

indication of how a professional, and in this case the professional is me.

How we go about it and how we, tacitly, and when we're doing this, this part of

our every day work, when we're grading work.

whether, whether we're grading student's work or we're evaluating proofs of other

mathematicians. This is all part of the tacit process of

grading. and, and in writing out a proof, I've

just, tried to isolate the things that I implicitly and automatically look at in

grading proofs. as indeed does, does any professor,

professional mathematician when they're grading a proof, okay?

and so, and what I'm really saying is, this is.

Not really perfect in terms of giving reasons.

arguably this one is more important than that one, for example.

You know, it's overall, however, I'm going to get four.

because these are sort of niggling. and the reason they're niggling is the

person has laid out the fact that this induction, has made it clear that they're

assuming it for n, have proved it for n plus 1.

they've definitely stated the use of the induction hypothesis.

as I said, this one would have been nice, I would have liked to see it, but this

author. Presumably decided it really was, was,

was patently obvious what's going on. this I think is an important one, because

that's critical to the proof. this is not something that's typical.

The farther you can pull this in depends on the, where the Fibonacci sequence is

defined, and this is the only part in the proof where we make use of that fact.

So you really should Should mention this one.

So this ones important, and this one's important.

And the reason is until we've got to the last line, all we've really done is we've

shown that the thing is true for an equals one, two.

Actually, further equals one, use the further and F2 was what it was.

So if there were few, they were put through for the first case, or observed

in the first case. And then we've shown that if it holds

stage n, it follows, and it holds stage n plus one.

So we've proved two simple facts. One, the fact about the first one, well,

it's the first two for that matter. And secondly, we've proved an implication

from n to n plus one. The conclusion is that this holds for

infinitely many number, for all of the actual numbers.

So somehow proving, two statements, one simple observation and one implication,

has proved that something is true for the infinitude of all natural numbers.

Now admittedly, induction has a natural it's always an obvious thing, this sort

of a, self evident truth to the principle of induction, you know.

You can think of it in terms of, of knocking rows of dominoes over or

something. So But the fact that the, that the, that

the principal of introduction, or the part, the method of induction has got a

sort of intuitive obviousness to it. Shouldn't obscure the fact that this is

actually a deep result. Making a conclusion about an infinite set

is non trivial, you know? The, the Hilbert Hotel tells us that

infinity's a very, paradoxical domain. We've got to be very careful.

So, this actually holds. We can make the conclusion that it's true

for all n. Because of the principle of mathematical

induction. This, in other words, this is a big deal.

This is a big, big deal. And when big deals are involved, you

should mention them. You know, if there's a big guy in the

room. it's polite, if not, [LAUGH] a matter of

self preservation to observe that fact and make it clear.

So you really do need to state the principle of induction here to state that

it's been used, or at least to say, by induction.

I, I, you know, if I was feeling, if it wasn't for the fact that the rest of the

proof was laid out so nicely. I might well have just deducted more

here. But as it is given everything else was

laid out so well and given that the proof is nice and elegant.

And this is a slick proof, there's almost no superfluous lines in.

And I think overall simply deducting two marks is about right.

So I've got 22 out of 24 for this one. and I feel reasonably good about this.

you know, the, couple of a small points, [COUGH] well, it's one small point, one

moderate big points, one really huge point, I think.

I think, this is being generous. But I think this is proof deserves

generosity. overall, I'm happy with that.

And, and this is really how it, it splits up.

Okay, let's go ahead and look at number five.

And number five is another of these Fibonacci sequence results.

This one says that if we take the first n Fibonacci numbers, add them together, the

result is the the next but one. Fibonacci number, we skip over Fn plus

one, we go to Fn plus two. Okay, so, okay, so as is typical for

these results it is proved by induction. So see how the proof goes.

For n equals one, the left-hand side is F1, that's why you get if n equals one,

there's no sum, it's just F1 itself. And the right-hand side is uh-oh, oh

dear. If n equals one, the right-hand side

would be F3 and F3 equals two, and one does not equal two.

So this isn't even true, it's not valid for n equals one, which means the

theorem's not valid. Oh, good grief, this is such an obvious

mistake. It's the kind of mistake that anyone

could make. Doesn't really reflect on their ability

as a mathematician. It's just a human error.

So common this kind of thing, and yet this is mathematics.

Ultimately in mathematics things are right or wrong.

I mean, you know, if, if this mathematics being used by an engineer to build a

bridge, and the bridge falls down and people get killed.

You know, that, that engineer could be held liable.

i am just defining or so you know at the end of the day we call that thing goes to

false ,i am have to 0 for logical correctness before, we are go further ,i

mean it is just a plan to evolve false result you could make the result correct

,you could make first result correct by subtracting one And it fact it turns out

as, and I'll come back to this, that if you put a minus 1 in here, then the

identity is true for all n. So there is a theorem here, and the get,

we get at the theorem by noticing what went wrong with this proof.

Incidentally, this is very typical in mathematics.

often in mathematics, the statement of the theorem when it's proved isn't the

one that the author originally tried to, to do.

Very often in mathematics we, we make a conjecture.

We try to prove it. the proof has gone wrong and so by

analyzing the proof we've thought of go back and change the statement.

So it's often the case that statements of theorems actually come after the proof.