Prove that if the integers 1, 2, 3, . . . , 65 are arranged in...

Prove that if the integers 1, 2, 3, . . . , 65 are arranged in any order, then it is possible to look either left to right or right to left through the list and find nine numbers that are in increasing order

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Colby Messinger answered 2 years ago

prove 2 is a factor of (n+1)(n+2) for all positive integers

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Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for every n ∈ N. That is, the sum of the first n perfect cubes is the square of the sum of the first n natural numbers. (As a student, I found it very surprising that the sum of the first n perfect cubes was always a perfect square at all.)

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Prove that there exists integers m and n such that 15m + 12n = 3 Please...

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