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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.

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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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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.

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