Question

Question 4 10 marks You should be able to answer this question after studying Unit 9. (a) Use proof by contraposition to prove that the following statement is true for all integers n: If n2 + 2n + 7 is odd, then n is even. [6] (b) Hence prove that the following statement is true for all integers n: na + 2n + 7 is odd if and only if n is even. [4]

Answer 1

Here given n²+2n+7 is odd. we know that 7 is odd number. Even number always divisible by 2. So 2n is divisible by 2. That implies an is even. Sum of One Odd and one even number is always Odd. Then to make m2 +2n+7 Odd. n2 must be a even number, because sum of odd & even number is oddi gp 2 is even n is even because even number square always even and Odd number square is al always Odd. So, this implics n is even." If n²+ 2n+7 is odd , then n is even. go

- b Question a proved n2+ 2n+7 is odd, then n is even. Now proof that If n is even then n2 + 2n+7 is odd. Here n is even then square of even number is even: Implies n2 even 2n divisible by 2 . So 2n is even. We know that 7 is Odd. So, Sum of two even number and One odd number is always odd. It means n2+2n+7 is Odd. So, we proved that n2+ 2n+7 is Odd it and only it n is even.
The conversely proof of (a) is (b)... both a and b make the 2nd statement true...