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Steps on solving 4 discrete math questions

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Please see attached file with questions and provide ALL work so that I can understand how you arrived at the answers. Thank you.


1. Let P (n) be the statement that 13+ 23+···+ n3 =(n(n+ 1)/2)2 for the positive integer n.
a) What is the statement P (1)?

b) Show that P (1) is true, completing the basis step of the proof.

c) What is the inductive hypothesis?

d) What do you need to prove in the inductive step?

e) Complete the inductive step, identifying where you use the inductive hypothesis.

e) Explain why these steps show that this formula is true whenever n is a positive integer.

2. Show ALL work. Solve the following systems of equations using Gaussian Elimination. Your procedure should be in matrix form.
x1 + x2 + x3= 1
2x1 - x2 + x3= 2
-1x2 + x3= 1

3. Determine the solutions of the system of equations whose matrix is row equivalent to
Give three examples of the solutions. Verify that your solutions satisfy the original system of equations.

4. Solve for B: B=

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Solution Summary

The solution gives detailed step on solving 4 discrete math questions on the topic of mathematical induction, Gaussian elimation and matrix operation. All formula and calculations are shown and explained.

See Also This Related BrainMass Solution

Discrete Math:Recursion

Discussion Questions

1. List all the steps used to search for 9 in the sequence 1,3, 4, 5, 6, 8, 9, 11 using a binary search.
2. Describe an induction process. How does induction process differ from a process of simple repetition?
3. Describe a favorite recreational activity in terms of its iterative components, such as solving a crossword or Sudoku puzzle or playing a game of chess or backgammon. Also, mention any recursive elements that occur.
4. Describe a situation in your professional or personal life when recursion, or at least the principle of recursion, played a role in accomplishing a task, such as a large chore that could be decomposed into smaller chunks that were easier to handle separately, but still had the semblance of the overall task. Did you track the completion of this task in any way to ensure that no pieces were left undone, much like an algorithm keeps placeholders to trace a way back from a recursive trajectory? If so, how did you do it? If not, why did you not?
Side Note: Here is the reason we discuss recursion for an IT degree. Recursion is the basis of searching and sorting...here is a quote from the following website (http://www.sparknotes.com/cs/recursion/examples/summary.html)
Next we'll look at how recursion can be used in searching and in sorting to increase the efficiency of these operations. Then we'll look at how recursion can be used for certain mathematical problems, such as printing a number in different bases and computing different sequences of numbers. For most of these problems, recursion presents an incredibly elegant solution that is easy to code and simple to understand.
Recursion may be difficult to understand at first, but once you do it becomes a powerful programming tool.
5. Given this recursive algorithm for computing a factorial...
procedure factorial(n: nonnegative integer)
if n = 0 then return 1
else return n *factorial(n − 1)
{output is n!}
Show all the steps used to find 5!

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