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Solving a mathematical problem about proposition and proof The following is the definition of "mysterious number": a positive integer that can be expressed as the square difference of two consecutive even numbers is called a mysterious number. Please judge 4, 12, 20, 28, 2012 as a mysterious number according to the definition of "mysterious number". Why?
(2n-2) ^ 2 - (2n) ^ 2 = 8N + 4 = 4 (2n + 1) is a mysterious number, which is an odd multiple of 4
So 12, 28, 2012 are mysterious numbers
(1 / 2 + 2 / 3 + 3 / 4 + 4 / 5 + 5 / 6 + 6 / 7) + (1 / 2 + 2 / 3 + 3 / 4 + 4 / 5 + 5 / 6 + 6 / 7) * 1 / 2 - (1 + 1 / 2 + 2 / 3 + 3 / 4 + 4 / 5 + 5 / 6 + 6 / 7) * (2 / 3 + 3 / 4 + 4 / 5 + 5 / 6 + 6 / 7)
(1/2＋2/3＋3/4＋4/5＋5/6＋6/7)＋(1/2＋2/3＋3/4＋4/5＋5/6＋6/7)×1/2－(1＋1/2＋2/3＋3/4＋4/5＋5/6＋6/7)×(2/3＋3/4＋4/5＋5/6＋6/7)
＝(1/2＋2/3＋3/4＋4/5＋5/6＋6/7)×(1＋1/2)－[(1＋1/2)＋(2/3＋3/4＋4/5＋5/6＋6/)]×(2/3＋3/4＋4/5＋5/6＋6/7)
＝(3/2)×(1/2＋2/3＋3/4＋4/5＋5/6＋6/7)－(3/2)×(2/3＋3/4＋4/5＋5/6＋6/7)－(2/3＋3/4＋4/5＋5/6＋6/7)×(2/3＋3/4＋4/5＋5/6＋6/7)
＝(3/2)×[(1/2＋2/3＋3/4＋4/5＋5/6＋6/7)－(2/3＋3/4＋4/5＋5/6＋6/)]－(2/3＋3/4＋4/5＋5/6＋6/7)^2
＝(3/2)×(1/2)－(2/3＋3/4＋4/5＋5/6＋6/7)^2
＝3/4－(2/3＋3/4＋4/5＋5/6＋6/7)^2
2/3＋3/4＋4/5＋5/6＋6/7
＝(1－1/3)＋(1－1/4)＋(1－1/5)＋(1－1/6)＋(1－1/7)
＝5－(1/3＋1/4＋1/5＋1/6＋1/7)
The least common multiple of 3,4,5,6,7 is 4 × 3 × 5 × 7 = 420
1/3＋1/4＋1/5＋1/6＋1/7
＝(4×5×7＋3×5×7＋4×3×7＋2×5×7＋4×3×5)/420
＝(140＋105＋84＋70＋60)/420
＝459/420
＝153/140
2/3＋3/4＋4/5＋5/6＋6/7＝5－153/140＝547/140
Original formula = 3 / 4 - (547 / 140) ^ 2 = - 284509 / 19600 ≈ - 14.51576531
Let u = {positive integer no more than 5}, a = {x | x2-5x + q = 0}, B = {x | x2 + PX + 12 = 0}, (∁ UA) ∪ B = {1, 3, 4, 5}, find P, Q and set a, B
The complete set u = {1,2,3,4,5}, a = {x | x2-5x + q = 0}, B = {x | x2 + PX + 12 = 0}, (∁ UA) ∪ B = {1,3,4,5}, ∪ 2 ∈ a, substituting x = 2 into x2-5x + q = 0, we get: 4-10 + q = 0, namely q = 6, namely x2-5x + 6 = 0, ∁ X-2) (x-3) = 0, namely x = 2 or x = 3, ∁ a = {2,3}, ∁ UA = {1,4
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