III. Explain why or why not the proof of the following result is correct or not correct. Let n be and integer. If 3n – 8 is odd, then n is odd. Proof. Assume that n is odd. Then n = 2k + 1 for some integer k. Then 3(2k + 1) – 8 6k + 3 – 8 6k – 5 2(3k – 3) + 1. %3D 3n – 8 = Since 3k – 3 is an integer, 3n – 8 is odd. Q.E.D.

Q: (a) Suppose that n > 1 is an integer with prime factorization n = pf"p.p. Prove that (k, + 1)(k, +…

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A: Mathematical induction

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Q: 12. What is the coefficient of x³y® in the expansion of (-3x + 5y)*? a) C(13,5) · 38 . 55 b)…

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Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

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Q: 1- Let n and r be positive integers such that r≤n. Then (a) P = nx P-1 (c) P = pn a O b O d n-1 (b)…

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Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

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Q: 3k k (3) Let n e N. Prove by mathematical induction that 4". k=0

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Q: c) n° n° + 2n is a multiple 3 for all n 2 1.

A: To show: n3+2n is a multiple 3 for all n≥1. Statement used: Principal of Mathematical Induction:…

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A: here is the proof

Q: 1. If n is an integer, then 2n +1 is odd. 2. The square of any integer can be written in the form 3k…

A: 1) Given:     If n is an integer, then 2n+1 is odd.   Let n=1  then 2n=21              =2

Q: B. Using induction, Prove that 2 + 6 + 10 + ... + (4n - 2) = 2n2 for all n e N. %3D

A:

Q: 1 + 2n < n° for all integers n² 2

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Q: 1- Let n and r be positive integers such that r≤n. Then (a) Pn=nx pn-1 (c) P = pn n-1 (b) P = P-1…

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Q: 2. Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n. for n ≥ 3, an = 2an-1+ bn-1…

A: NOTE:- AS POSTED MULTIPLE INDEPENDENT QUESTIONS. WE ARE ANSWERING ONLY FIRST QUESTIONS FOR YOU AS…

Q: #2. The binomial theorem states that for any real numbers a and b, In (a + b)n = =Σ" ("a²-kbk for…

A:

Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

A:

Q: (-1)"(n + 2) 2n+1(n + 1) Compute the exact value of n=0

A: The ratio test can be used to assess if a series is converging or diverging. The following are the…

Q: Prove by, Mathematical Induction (a)Prove that 12+ 22+ · · · + n2 =1/6 n(n + 1)(2n + 1) for all n ∈…

A: Prove by, Mathematical Induction (a)Prove that 12+ 22+ · · · + n2 =1/6 n(n + 1)(2n + 1) for all n ∈…

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A: Given:

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Q: 8. If n+1, 2n-1 and n+5 are in AP. Find the value of n

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Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

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Q: [Hint: For m > 2, ( 2 7. For n > 1, verify that + 1 1² + 3² + 5² + ... + (2n – 1)° = ( 2"" ) - 8.…

A: Let's say we need to prove a formula or a statement P(n). To prove this, we need to prove that The…

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Q: Show that (m) (2n 4n n

A: As n → ∞, By Striling approximation formula n! tends to  sqrt(2 pie n) (n/e)^n

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A: It is clear that from N(A)

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A: Solution : We have given the expression 4 + 8 + 12 + … + 4n = 2n(n + 1) and We need to prove the…

Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

A:

Q: 10. Prove that the sum of n squares can be found as follows n(n + 1)(2n + 1) 12 + 22 + 32+...+n² : 6

A: I use  a3-b3= (a-b)(a2+ab+b2)

Q: 7. (a + b)"= E=o( n choose k) a™-kb* for any integer n>=0. Use this theorem to compute (2x-3)* ) The…

A: Given, 2x-34

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A:

Q: 11) Which of the following shows the solution for n2- 2n < 24? 十十 A) 6-54-3-2-10123456 + +++++++…

A:

Q: 10. Prove that the sum of n squares can be found as follows 12 + 22 + 3°+...+n² n(n + 1)(2n + 1) 6.

A:

Q: Use Gauss’s trick to determine H(n) = 1 + 5 + 9 + 13 + … + (4n – 7) + (4n – 3) + (4n + 1) I'm…

A: Explained below

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A:

Q: Find all m, n E N such that: n! + 2n + 49 = m²

A: This problem can be solved by graphical approach. Use a graphing utility to plot the graph of…

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A:

Q: 2. The square of any integer can be written in the form 3k or 3k + 1, for some k E Z.

A: True. We use the fact any integer is of the form 3q,3q+1,3q+2.