So I can just go straight in and

put the numbers 4 in everywhere and a total of 24 for this one.

And this is very much like the solution we saw in sample exam solution two,

except in that case, the answer had an elementary mathematical mistake,

here an arithmetical mistake where the student put 8 rather than 10.

But other than that, this is essentially the same argument.

It's written extremely slickly and

everything's there that you need to be there.

I wouldn't normally expect a typical student in the class to produce something

quite as short and succinct as this, but it's absolutely a perfect proof.

All of the reasoning is there, everything you need there, so very nice little proof.

Looking at number 3 now, I'm not going to put in all of the 4s anymore,

I'll just jump straight in and put the 24, a beautiful proof.

Consider the two cases even and odd separately, very elegant.

If it's even, then that guy works out as being odd and

I've actually stated that it's odd.

I admit that I was the one that wrote this out.

If n is odd, same thing, you follow it through and you show that this guy is odd.

And then you wrap this thing up by concluding that in both cases n squared

+ n + 1 is odd.

And the danger with this particular video is that it's going to go by too

quickly for you to sort of take it in.

So what I suggest you do is as we get to each one you freeze the video and

you just take a look at it and you see why it works.

Incidentally, some of the early research that was done at Stanford onto students

doing MOOCs showed that there was a very strong correlation between the number of

times the students used the pause and rewind button and

the success they had in the course in the long run.

And that's for a whole variety of reasons.

One is, of course, it shows that the student's engaged,

if you have to keep stopping and rewinding.

It shows you're engaged, it shows you're reflecting on the work,

it shows you understand when you don't understand and what's going on.

You know when you don't understand.

And you're taking control of your own learning.

So the correlation between using the pause and rewind button and

success in the course is actually quite high.

And indeed one of the advantages that MOOCs have over classroom lectures

is that you can do that.

It's very hard when I'm lecturing in the class to freeze me and rewind me.

In fact, I wouldn't recommend you try to do that, okay?

I certainly hope you wouldn't try anyway.

Okay, let's go on to number 4.