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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:
  1. \begin{align*}- 1 - 5 - 9 + ... - 4k + 3 = k(-2k + 1)\end{align*}159+...4k+3=k(2k+1)
  2. \begin{align*}1 + 4 + 4^2 + 4^3 + ... + 4^k = \frac{4^{k+1} - 1}{4 - 1}\end{align*}1+4+42+43+...+4k=4k+1141
  3. \begin{align*}3 + 6 + 9 + ... + 3k = \frac{3}{2} k(k+ 1)\end{align*}3+6+9+...+3k=32k(k+1)
  4. \begin{align*}-1 - 3 - 5 + ... - 2k + 1 = k(-k)\end{align*}135+...2k+1=k(k)
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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: \begin{align*}1^3 + 2^3 + 3^3 + ... + n^3 = \frac{n^2(n + 1)^2}{4}\end{align*}13+23+33+...+n3=n2(n+1)24
  3. Proove: \begin{align*}n^2 \frac{(n + 1)^2}{4} = 1^3 + 2^3 + 3^3 + ... + n^3\end{align*}n2(n+1)24=13+23+33+...+n3
  4. Prove that 3 is a factor of \begin{align*}n^3 + 2n\end{align*}n3+2n 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: \begin{align*}x_1, ..., x_n\end{align*}x1,...,xn are positive numbers, prove the following: \begin{align*}\frac{(x_1+ ...+ x_n)}{n} \geq (x_1 \cdot ...\cdot x_n)^{\frac{1}{n}}\end{align*}(x1+...+xn)n(x1...xn)1n
  3. \begin{align*}n! \geq 3^n\end{align*} for \begin{align*}n = 7, 8, 9, ....\end{align*}

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