標題:
F.4 maths~~~15marks
發問:
1. Prove that the sum of three consecutive numbers is a multiple of 3. 2. Prove that the sum of four consecutive even numbers cannot be 56 3. Prove that the sum of three consecutive odd numbers is a multiple of 3. *Please list the detailed steps.
最佳解答:
1. Let the first number be x, where x is integer So, the second number is x+1; third number is x + 2 Then, sum = x + (x + 1) + (x + 2) = 3x + 3 = 3(x+1) which is multiple of 3, where the quotient is x+1 2. Let the first number be 2x , where x is an integer. So, the second, third, and fourth number should be: 2x+2, 2x+4, 2x+6 sum = 2x + (2x+2) + (2x+4) + (2x+6) = 8x + 12 Assume sum = 8x + 12 = 56, then 8x = 44 x = 5.5 which is non-integer But x is integer. So, sum can not be 56. 3. Let the first number be 2x+1 , where x is integer then, the second and third number should be 2x+3 and 2x+5 respectively. sum = (2x+1) + (2x+3) + (2x+5) = 6x+9 = 3(2x+3) which is multiple of 3. 2009-08-21 23:59:37 補充： For question 2), we have applied the set theory: If P is true, then Q is true. In other words, If Q is false, then P is false. That is, the statement P -> Q is equivalent to the statement ~Q -> ~P 2009-08-21 23:59:47 補充： P refers to the statement : x is integer Q refers to the statement : sum of four consecutive even numbers = 8x + 12 != 56 That is, prove that if P is true, then Q is true. 2009-08-21 23:59:52 補充： To prove that 8x+12 != 56 if x is integer, we just need to show that x is non-integer if 8x + 12 = 56 That is, show that ~P is true is ~Q is true. 2009-08-22 00:01:45 補充： Sorry... Typing mistake: referring to 2009-08-22 00:00:02 補充 : To prove that 8x+12 != 56 if x is integer, we just need to show that x is non-integer if 8x + 12 = 56 That is, show that ~P is true is ~Q is true. The last statement should be: That is, show that ~P is true if ~Q is true.
F.4 maths~~~15marks
發問:
1. Prove that the sum of three consecutive numbers is a multiple of 3. 2. Prove that the sum of four consecutive even numbers cannot be 56 3. Prove that the sum of three consecutive odd numbers is a multiple of 3. *Please list the detailed steps.
最佳解答:
1. Let the first number be x, where x is integer So, the second number is x+1; third number is x + 2 Then, sum = x + (x + 1) + (x + 2) = 3x + 3 = 3(x+1) which is multiple of 3, where the quotient is x+1 2. Let the first number be 2x , where x is an integer. So, the second, third, and fourth number should be: 2x+2, 2x+4, 2x+6 sum = 2x + (2x+2) + (2x+4) + (2x+6) = 8x + 12 Assume sum = 8x + 12 = 56, then 8x = 44 x = 5.5 which is non-integer But x is integer. So, sum can not be 56. 3. Let the first number be 2x+1 , where x is integer then, the second and third number should be 2x+3 and 2x+5 respectively. sum = (2x+1) + (2x+3) + (2x+5) = 6x+9 = 3(2x+3) which is multiple of 3. 2009-08-21 23:59:37 補充： For question 2), we have applied the set theory: If P is true, then Q is true. In other words, If Q is false, then P is false. That is, the statement P -> Q is equivalent to the statement ~Q -> ~P 2009-08-21 23:59:47 補充： P refers to the statement : x is integer Q refers to the statement : sum of four consecutive even numbers = 8x + 12 != 56 That is, prove that if P is true, then Q is true. 2009-08-21 23:59:52 補充： To prove that 8x+12 != 56 if x is integer, we just need to show that x is non-integer if 8x + 12 = 56 That is, show that ~P is true is ~Q is true. 2009-08-22 00:01:45 補充： Sorry... Typing mistake: referring to 2009-08-22 00:00:02 補充 : To prove that 8x+12 != 56 if x is integer, we just need to show that x is non-integer if 8x + 12 = 56 That is, show that ~P is true is ~Q is true. The last statement should be: That is, show that ~P is true if ~Q is true.
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