Further Proof & Reasoning (DP IB Maths: AA HL)

Prove that the equation  has distinct real solutions for all values of , where .

Prove by mathematical induction that  is divisible by 16.

Prove by contradiction that  is irrational.

Prove by exhaustion that the sum of two consecutive square numbers between 100 and 200 is an odd number.

The three statements below are false.

In each case verify the statement is false by use of a counter example and state an alternative domain that would make the statement true. 

(i)

 

(ii)

is a prime number for . 

(iii)

Use mathematical induction to prove that the th derivative of the function  is  given by

for all integers, , where .

Prove that  if .

The product of three consecutive integers is added to the middle integer. 

Prove that the result is a perfect cube.

Prove by mathematical induction, that for ,

Use a contradiction to prove that the difference between a rational number and an irrational number is irrational.

Prove that there are no non-zero real values of  and such that  .

Prove by mathematical induction that if  then  .

Prove by mathematical induction that

Prove that there are no real values of  such that the equation has no real solutions.

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 .