WJEC Mathematics for AS Level - Pure

1 Proof 14 3 Prove the following proposition: If a , b are positive real numbers, then a 2 + b 2 ≥ 2 ab . Answer 3 Assuming that positive real numbers a , b exist such that a + b ≥ 2 √ ab a 2 + b 2 ≥ 2 ab ( a + b ) 2 − 2 ab ≥ 2 ab ( a + b ) 2 ≥ 4 ab a 2 + 2 ab + b 2 ≥ 4 ab a 2 − 2 ab + b 2 ≥ 0 a 2 + b 2 ≥ 2 ab ( a − b ) 2 ≥ 0 Hence the original proposition that a + b ≥ 2 √ ab is true. Examination questions may not always mention the method of proof to be used so you will have to decide the method to use. • If there are a limited and small number of values to try in order to establish whether a statement or conjecture is true or false, use proof by exhaustion. • If it looks like a proof that can be solved algebraically, use proof by deduction. • If the proof has an inϐinite number of values to test, then try to ϐind values that do not work and use disproof by counter-example. BOOST Grade ⇪⇪⇪⇪ Step by Here is a statement: ‘Every integer that is a perfect cube is a multiple of 9 or one more than a multiple of 9 or one less than a multiple of 9’. Steps to take 1 There are three different proofs we could use so we need to select the one to use. 2 As there are three different cases to try (i.e. cube is a multiple of 9 or one more than a multiple of 9 or one less than a multiple of 9) we can use proof by exhaustion. 3 Think about how to form three equations that can be used to test the three cases. 4 Think about what letters you will use for the values that vary (e.g. n can be the integer you start with). Answer Each cube number is a cube of some integer n . All integers can be described as either a multiple of 3 or one less or two less than a multiple of 3. Squaring both sides removes the square root from the right-hand side of the equation. Note this will always be true if a and b are both real numbers, because when squaring a real number, you always obtain a number ≥ 0. If you are still having trouble getting to grips with the different proofs, take a look at some of the videos on YouTube which explain the three proofs: Proof by exhaustion, Disproof by counter-example, Proof by deduction. Try to ϐind some which only apply to AS-level. Active Learning