Prove that, for all integers n ≥ 0 and 0 ≤ k ≤n, we have (x+¹)=()+(*+¹) ++ (1) n+ k+1 k Hint: the left hand side is the number of k + 1-element subsets of {1,..., n+1}. Show that the right hand side is also equal to this number by grouping these subsets according to their largest element.

Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

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Q: Determine the least upper bou You must fully justify your an

A: We have to find least upper bound of the given set

Q: 1) In Exercise 4.78, it was stated for every two positive real numbers a and b that + > 2. Use this…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Prove for all integers n ≥ 1 that 1(2) +2(3) +3(4) + ... + n(n + 1) = n(n+1)(n+2) 3

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Q: (2n- 2. Give a combinatorial argument to show that k- ways the number of ways to select a committee…

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Q: ) Prove every integer n >= 7, n! > 3n

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Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: Show that (n) ‡ 14 for all n ≥ 1.

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Q: 13. Prove that 6 | (n³-n) for every integer n ≥ 0.

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Q: (1) Let S be the smallest set of positive integers such that (a) 2 is in S, (b) n is in S whenever…

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Q: Prove: If m and n are non-zero integers and m > n, then there exists an integer q such that (1/n -…

A: Let m and n are non-zero integers such that m>n To prove there exists an integer q such that…

Q: Prove for every integer n > 8 that there exist nonnegative integers a and b such that n = 3a + 5b.

A: Proof by induction For the base case: n=8, we have 8=5+3. Suppose that the statement holds for k…

Q: Compute the infimum and supremum of the set {n+(1/n)" : n E N} if they exist.

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Q: 8. Let K be an integer that is greater than 1, a perfect square, and equal to Ei for some integer D.…

A: A perfect square is a positive integer whose square root is a positive integer. Example of perfect…

Q: Prove that k (x)= - = n (n − 1) for integers n and k with 1 ≤ k ≤ n, using a: a) algebraic proof.

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Q: Let P(n) be the statement n2 ≤ n! where n is a nonnegative integer. For which nonnegative integers…

A: check if the result is true for n=1 n2≤n!12≤1(true) for n=2 22≤2!4≤2(false) for n=3  32≤3!9≤6(false)…

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

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Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  => k | {(n+4) - (n-1)} => k | 5

Q: Compute the supremum and infimum of the set A = {n/(2n+1) : n ∈ ℕ}

A: We have to find the supremum and infimum of the following set A =n(2n+1):n∈ℕ

Q: Prove that 31 (n³ +5n+6) for every integer n ≥ 0.

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Q: Prove that for every positive integer n the number (1 – /2)" is not ratioanal.

A: Given: The given number is, (1-2)n where n is the positive integer.   To prove: The number (1-2)n is…