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Induction and Factors

Proving divisibility using properties of integers and inductive proofs.

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Inductive Proofs

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Vocabulary

Complete the chart.
Word Definition
Mathematical Induction ________________________________________________________
______________ the total of all of the numbers in a series
nth term ________________________________________________________
______________ a number or an expression that is multiplied with other factors to create a product
______________ the product of the positive integers from 1 to some value n: n! = 1 × 2 × 3 × 4...× (n-1) × n
Inequality ________________________________________________________
Postulate ________________________________________________________

Inductive Proofs

Steps of Mathematical Induction:

Step 1) The base case: prove that the statement is true for __________________. Often with induction you may want to expand the first step by showing that the statement is true for several _______________.

Step 2) The inductive hypothesis: assume that the statement is true for ________________. 

Step 3) The inductive step: use the inductive hypothesis to show that the statement is true for ______________.

Apply these steps to the integer sum formula:

Step 1) _______________________________________________________________

Step 2) _______________________________________________________________

Step 3) _______________________________________________________________

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Use induction to prove the following:
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Induction and Factors

Properties of Integers and their Factors

Complete the properties and prove with induction.

Property 1 : If is a factor of , and is a factor of , then is a factor of _____________.

Proof: _________________________________________________________

Property 2 : If is a factor of and is a factor of , then is a factor of ____________.

Proof: _________________________________________________________

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  1. Without adding, determine if 8 a factor of 56 + 80
  2. Proove: 
  3. Proove: 
  4. Prove that 3 is a factor of  for all positive integers .
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Induction and Inequalities

Describe the following properties of inequalities:

Transitive Property: _________________________________________________

Addition Property: _________________________________________________

Multiplication Property: _________________________________________________

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Prove the following inequalities:

  1. The side length of a pentagon is less than the sum of all its other side lengths.
  2. Given:  are positive numbers, prove the following: 
  3.  for 

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