Question

In: Advanced Math

3. To begin a proof by contradiction for “If n is even then n+1 is odd,”...

3. To begin a proof by contradiction for “If n is even then n+1 is odd,” what would you “assume true?

4. Prove that the following is not true by finding a counterexample.

“The sum of any 3 consecutive integers is even"

5. Show a Proof by exhaustion for the following: For n = 2, 4, 6, n²-1 is odd

6. Show an informal Direct Proof for “The sum of 2 even integers is even.”

Recursive Definitions

7. The Fibonacci Sequence is defined as follows:

F(1) = 1

F(2) = 1

F(n) = F(n-1) + F(n-2) for n>2.

The first 10 numbers in the sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

Find F(13) = ______________ and F(14) =______________

8. Prove the following property of the Fibonacci numbers directly from the definition.

F(n+3) = 2F(n+1) + F(n-3)+ 2F(n-2)

9. Find the 3^rd, 4^th and 5^th values in the sequence.

D(1) = 0

D(n) = 2D(n-1) + 2 for n>1.

D(2) = _______________________________________________

D(3) = _______________________________________________

D(4) = _______________________________________________

10. Find the 3^rd, 4^th and 5^th values in the sequence.

D(1) = 1

D(2) = 4

D(n) = 2D(n-1) + D(n-2) for n>2.

D(3) = _______________________________________________

D(4) = _______________________________________________

D(5) = _______________________________________________

11. Using the SelectionSort algorithm on p 169 in your textbook, simulate the execution of the algorithm on the following list L; write the list after every exchange.

L: 9, 2, 4, 8, 6

_______________

Expert Solution

Colby Messinger answered 1 month ago

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