1.5 Further Proof & Reasoning

Using the method of proof by contradiction, prove that  is irrational.

Using mathematical induction, prove that  is divisible by 5 for .

The th triangular number is given by the formula .

Write down the first five triangular numbers.

Prove by exhaustion that the first five triangular numbers are all factors of 180.

Determine, with appropriate reasoning, whether the following statements are true or false: 

(i)

Given and  is divisible by 4, then  is divisible by 4.

(ii)

Given then  is a prime number.

(iii)

Given and  is divisible by 3, then  is divisible by 3.

Prove that  is positive for all real values of .   

Show that , where .

Hence, or otherwise, prove that is a multiple of 8.

Prove that .

Prove that

Prove by mathematical induction  given 

Prove by mathematical induction that if   then 

Prove by induction that

for all values of .

Given 

(i)

prove that ,

(ii)

prove that, for ,

Prove that is divisible by 5 for .

Prove that there are an infinite number of prime numbers.

523 and 541 are prime numbers.

Prove by exhaustion that these are consecutive prime numbers.

Three of the four statements below are false.

Eliminate the false statements by providing a counter example and thus deduce the true statement.

(i)

(ii)

Every th triangular number is even, .

(iii)

.

Prove that the equation  has no integer solutions.

The function  is given as  where  is an integer.

Find and .

Prove that is not prime for all values of .

Write down and  from smallest to largest, given and  and 

Write down and  from smallest to largest, given  and

Prove ,  given .

Show that the derivative of  is

.

Prove, by mathematical induction, that for

Given that the graph of   touches the -axis at the point with coordinates , prove that  for all real values of .

Prove that ,  for all   .

Proof by Induction that

for all positive integer values of .