Find n if: a) P(n, n) = 5040. b) 66= C(n, 2). c) P(n+1, 5)= 5P(n,1) Note: n is a nonnegative integer and parts (a), (b), and (c) are not connected.

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A: We have to proff the question

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A: To answer the given questions we shall use the following definition  An integer n is said to be even…

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A:

Q: Let n EU = Z and p(n): n² + n is an even integer. a. Write the following statement: (Vn) p(n). b. Is…

A: (a) Here, the given statement is: For ∀n∈U=ℤ, p(n):n2+n is an even integer.

Q: 1b) Given yadrilateral KLMN with K(-,4),L (2,2),Mlo-1) and NC-3,1). Verify that KLMN is a Square,…

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A: Here, we use rules of divisibility.

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A:

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Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4m + 3 бт + 1

A: We have to do proper manipulation.

Q: Q:- Show thet B(n, n) B( n+on+=2 1-40

A: The solution is given as

Q: Computer (73112). (Hint: you do not need to multiply out 7³11².)

A: We have to compute .We know thati) is the number of positive divisors of n .

Q: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that are…

Q: 1. Show there exists an integer n such that n² + 3n/2 = 1. Is the integer unique? Explain your…

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Q: "Co 34 "C, "C, "C; (-1)" "C, ,ne Nis equal to 10 14 (4n + 2) 2" n! 2"-n! () 3.5-7...(2n - 1)(2n +1)…

A: We have use binomial theorem  1-xn=nC0 - nC1 x + nC2 x2  + .........+(-1)n   nCn

Q: Find a positive integer n such that 2n is a square and 3n is a cube.

A: Sol:- Let 2n be equal to x², where x is a positive integer. Then, n can be written as n = x²/2.…

Q: 3) (n² + 10n+18) ÷ (n+ 5)

A: To divide the following expression.n2+10n+18÷n+5

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A:

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

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Q: (22) Prove: Sn: 1 ² + 2 ² + 3² + ... + n² = n(n+₁) (2nti) neN

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Q: Evaluate SA(u) du if A(u) a) 6i+8k b) -6i+8k c) 61-8k (3u²-1)i + (2u-3)j + (6u²-4u)k. d)-61-8k

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Q: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that…

Q: ) 6. Express the sum 9 + 28 + 65 + ... + (n'+1) using E-notation.

A: We have to find sum of series:

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Q: 17 3. Evaluate: de (r- 1)(2 + 16)

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Q: (15) Let a, b = Z. Prove that if a is divisible by 4 and b is divisible by 4, then: (a) ab is…

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Q: Any integer of the form 12k +11 is also of the form a) 3m + 1 b) 5m +2 4m + 3 6m + 1

A: Answer

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A: Introduction: Consider a starting point for which the assertion is correct. The proposition is true…

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Q: 9. 10. What is t divisors?

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Q: I need help

A: To find:∫013481-u221+u23du

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A: Set: A well-defined collection of objects or numbers is called a set. We can write a set in two…