A Level Maths Made Easy - Proofs

hey guys and welcome back to another a level math Made Easy video today we're going to be looking at proof but I want to just say full disclosure this is not going to include proof by contradiction because proof by contradiction is far more in depth in detail there's a lot more that they can ask you whereas this type of proof partly is just kind of GCSE sty of proof but also there's some new stuff that I wanted to go through just so you know it's the year 12 regular proofs that they can ask you so without further Ado let's just jump straight into the first question first question just asks you to prove prove that x^2 - 4x + 7 is positive for all values of X and this is something that funly enough a few students aren't too sure about so when you do this you have to do it algebraically so the proof that we do for the most part in a level is going to be algebraically and that and the reason for that is if we do it using algebra so using just letters and variables we prove it for every single possible value if you think about this if I try and Sub in values I'd have to sub in an infinite number of values to show it's always positive right i' have to do from minus infinity to plus infinity instead if we do it through algebra you only have to do a few values now the question is how do you prove that a quadratic is always positive well if I write it like this it might help you you're trying to prove that this thing is always greater than zero the way I would do this and the way I one of the only ways that I can think of to do this is to just do completing the square because completing the square gives you the minimum point if that minimum point is above zero then you're done so if we do completing the square on this remember how to do that all you need to do is you half the number in front of X so we have x - 2^ 2 - - 2^ 2 and then + 7 expanding that you get x - 2 all 2 then - 4 + 7 and lastly you get x - 2^ 2 ++ 3 now how do I know that this is is always positive well you can get the minimum or technically maximum point from this all you do is you change the sign of the number in the bracket so instead of a minus two it's a plus two and you keep this sign exactly the same if you go back to the completing the square video I do explain how that works but essentially how it works is if you try and look for the smallest value this is you look at the part that can vary which is this bracket what is the smallest value this bracket can be well if it's negative it gets squared and becomes positive so the smallest value this can be is when the bracket is equal to zero which happens when x equal 2 that is the smallest thing this function can be overall now this is greater than zero right the Y value is greater than zero therefore always positive and that works for literally every value of x and I haven't had to sub in a billion different values the next thing I can ask you is and this is quite interesting is it's pretty much just the GCS style of question so prove and then some kind of statement now I want to point out that if you want even numbers then you're going to write it as 2 N if you want odd numbers it' be 2 n + 1 the word consecutive means that the numbers follow each other so what do I mean by that well the numbers one two are consecutive three and four five and six and so on consecutive odd numbers would be like this 1 3 5 7 and so on so the idea is they can ask you for consecutive odd or even numbers all you need to do is add two to both of these Expressions why because if you notice uh between every pair of odd numbers that are consecutive there's always a plus two and the same with even right the even numbers are 2 4 6 there's always going to be an add two in between them so all you need to do is take these original expressions for even and odd and add two to them in order to wipe this out so let's go through the motions now prove that the sum of two consecutive odd numbers so we have 2 n + 3 right because 2 n + 1 + 2 and then you have 2 N plus another two so plus 5 okay and we need to prove that it is a multiple of four now in this case if you want to prove that a expression is a multiple of anything with 2 4 8 million three whatever all you're going to do is try and factorize uh that number out so we're going to try and factorize a four out of this expression by the way an even number is a multiple of two so if they ask you to show that an expression is always even for example you're just going to try and factorize out a two so collecting the like terms we get 4 n Plus 8 which I can write as 4 n + 2 so because we have this uh four factorized out it must be a multiple of four and why is that well if you think about it if you do four time any integer you're always going to get a multiple of four right four * 1 4 * 2 4 * 3 and so on how do I know it's an integer because it's an odd number so this is an odd number well n is just a number plus two that's still an integer Time 4 still an integer and it also must be a multiple of four prove that the sum of the squares of two consecutive on inches is always two more than a multiple of eight again students kind of get into their own heads about this they get a bit too worked up break it down so it's so it's a sum so we're going to have an add squares so we're going to be squaring two things like this of two consecutive odd integers so again we're going to have that 2 n + 3 and 2 n + 5 and then it says it's always two more than a multiple of eight that part might seem a bit weird and it is weird but first of all if you just do the so first of all you get a mark for this if you actually just do the expansion simplifying you get most of the marks anyway so expanding this we would get 4 n^2 then you get + 12 n Plus + 9 then plus again 4 n^ 2 + 20 n + 25 so then we get 8^ 2 + 32n + 34 prove there always two more than a multiple of 8 that's a bit tricky here's how I would do it if you have a look at this this is divisible by8 this is divisible by8 but this isn't however if I to break this 34 into 32 + 2 that is now a multiple of 8 so what they're intending for you to do here is to do this oops how do I know that that that you can do that well to be honest I wouldn't have known to do that unless it told me it's always two more than so I'm trying to split up the 34 into two and another number it must be 32 then if I factorize this we get 8 n^2 + 4 n + 4 + 2 so this is always of course a multiple of eight and then this means it's two more so it's two more than a multiple of eight always so again that little trick there is a bit tricky it's not really something you would have known but you can kind of get the hint from the way the question is worded and here we're going through the last one one so I know this might seem a bit of a shorter one um I a bit ill so that's why I decided to pick a topic that was slightly shorter but again with the proofs it's all about just practicing as many as humanly possible there aren't too many tricks involved um and we've pretty much gone through quite a few of them so writing odd and even numbers consecutive numbers and of course showing that things are positive the last St I wanted to go through is proving these kind of inequalities as you can see it's still a quadratic so a smart thing to do would be to move everything over to one side because with a quadratic generally speaking you want to equal to zero or at least one side of the equation to equal zero so moving this over we get n^2 + 6 n + 12 and now we're actually just proving that this is positive which links back to the very first question we did in the video we're just going to use completing the square so again if you ever have a quadratic it's always good to make sure one side is equal to zero it is a very common uh step that I don't see many people take all the time um they sometimes get a bit flustered to get a bit in their own head so again doing the completing the square we have n uh + 3 squar 3 S in fact yeah we do this so n + 3 S - 9 + 12 and as you can see this is indeed bigger than zero if for some reason we got a negative number then we would know that this statement is actually false which kind of links to proof by contradiction which we'll be doing in a future video but that is essentially how you do these kinds of proofs really important to do because these are worth three or four marks which is quite a lot to be to be honest um but yeah we'll leave it there