11.6. Backward Reasoning

Try these first

1. 1.

   What is backward reasoning?

   Hint

   Think about how one might work backward from an answer.

2. 2.

   What does 'distinct real numbers' mean?

   Hint

   Consider the definition of two numbers not being equal.

3. 3.

   What does backward reasoning help to achieve in mathematical proofs?

   • Proves the conclusion directly
   • Identifies true premises
   • Confirms all variables

   Hint

   Think about starting from the conclusion.

4. 4.

   True or False: The arithmetic mean is always greater than or equal to geometric mean for distinct positive numbers.

   • True
   • False

   Hint

   Think of their definitions.

5. 5.

   Prove that for any two distinct integers, the difference of their squares is the product of their sum and difference using backward reasoning.

   Hint

   Investigate the factorization of the difference of squares.

6. 6.

   Use backward reasoning to prove that if one number is greater than another, then their squares retain this relationship if both are positive.

   Hint

   Think about positive numbers and their squares.