Examine the sum of the first n positive integers, E1 (2i – 1) = 1+3+5+...+(2n – 1) i=1 Provide a proof that this sum has the explicit formula of n^2. That is, prove that Vn e Z+, P(n): E (2i – 1) = n? =D1 Requirements: -Identify the basis step, the inductive step, and if any assumptions, deductions, and conclusions -Proof must include a mathematical statement on the with its justification on the right as such below

Q: QUESTION 23 There are 150 seats in the auditorium. 40 located on the left; 50 located on the right;…

A: At least 23 means 23 or 24 or 25.There are 60 seats which are located in the center. So (150-60)=90…

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A: No. of parking spots =5 No. of people =5

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A:

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A:

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Q: Explain each step of this proof that C(5, 5) + C(6, 5) + C(7, 5) = C(8, 6): Proof. C(5, 5) + C(6, 5)…

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A:

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A: Given that,

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A: There are n parking spots 1, 2, ..., n on a one-way street Cars 1, 2,..., n arrive in this order.…

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A: We have to prove that

Q: DISCRETE MATHEMATICS Part I. IDENTIFICATION.    V= _________________________  E=…

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A:

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A: Both A and B must be non-empty. So, possible cardinality of A can be  1,2,3, ..., n-1.

Q: max z = 3x1 + x2 s.t. 2x1 + x2 < 6 X1 + 3x2 < 9 X1, X2 2 ()

A:

Q: Give a combinatorial proof that: 2n 2+()²-(²-1) = n k n k n n k=1 [Hint: There are two groups of n…

A: Given is ∑ k=1nknk2=n2n-1n-1.    We have to prove it combinatorial.

Q: Exercise 2. Use a combinatorial argument to prove É(:)-("*") n+r+ k=0

A: ∑k=0rn+kk=n+r+1r LHS=n0+n+11+n+22+⋯+n+rr       =n0+n+10+n+11+n+22+⋯+n+rr-n+10       =…

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A:

Q: Find (AB) U (AC)  (AB)C  b) Prove the Boolean rule (A+B)(A+C) = A+BC

A: (1) AB={2,3,5,7}{2,4,6,8}={2,3,4,5,6,7,8} AC={2,3,5,7}{ }={2,3,5,7} (AB)U(AC)=(AB)={2,3,4,5,6,7,8}…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.