How many 2-permutations are there of a set of eight objects? Prove that for all integers n > 2, P(n + 1, 3) = n³ – n.

Q: Use induction to prove that the sum of all the odd integers 1, 3, 5, 7, n to any odd integer n is a…

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Q: let a and b be two relatively prime integers. if there exists an m such that (a/b)^m is an integers,…

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Q: If a set S has n > 3 elements, how many subsets of S have more than one element, but less than n…

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Q: Prove that if n is an even integer greater than 2, then 2 - 1 is a composite.

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Q: (Prove by induction that for all integers n ≥ 1, 1² + 32 +5² +...+(2n-1)² = n(2n-1)(2n + 1) 3

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Q: Prove that for all positive integers n, 13+23+33+…+n3=n(n+1)22.

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Q: Prove that 25|(6" – 5n – 1) for all positive integers n. -

A: We can prove given statement by principle of mathematical induction: Alogrithm: Obtain P(n) and…

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

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Q: 7. Ug = kuz-1¬- Ug–2, for every integer k > 3 uj = 1, u2 = 1

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Q: Prove that for any odd positive integer, n, the sum of nconsecutive positive integers is divisible…

A: To prove for any odd positive integer, n, the sum of nconsecutive positive integers is divisible by…

Q: Prove there are no natural numbers n and p with n ≥ 2 and p² + 1 = 2”.

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Q: Show that for every positive

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Q: 3. Prove that 2n > n² for all integers n such that n > 4.

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Q: Prove that for all n in J, 1 + 3 + 5 + • • • + (2n - 1) = n2

A: We have to prove 1+3+5+....+2n-1=n2

Q: An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of every…

A: Given a n×n magic square

Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

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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: Prove that 3" < n! for all integers n 2 7.

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Q: (16) Prove that 22n – 1 is divisible by 3 for all n € N.

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Q: Prove: Any positive integer can be represented as an aggregate of different powers of 3, the terms…

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