2015年7月15日 星期三

訓練數學感 70 ─ 格子點

http://4rdp.blogspot.tw/2015/07/70.html

坐標平面上,X 坐標與 Y 坐標皆為整數的點稱為「格子點」。設 n 為正整數,已知在第一象限且滿足 的格子點 (x,y) 的數目為 。則 的值為多少?


這題是 104年大學指考的一題,由網友行天下轉傳給我,好久沒算這麼高等的數學題。

5 則留言:

  1. 1 ?
    First, we have to find the relation between an and n.
    Second, find the limit.

    Result : an = (n+1) ^2
    an / n^2 = 1 + 2*n^-1 + n^-2
    as n tends to infinity, n^-1 and n^-2 tend to 0.
    Thus, the final result should be 1.

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    1. Hi 小文,

      Regarding the an, it isn't equal to (n+1)^2.

      It should be
      an = (n+1)(2n+1)/2 - n -(2n+1) = (2n^2 - 3n -1)/2

      The final result is equal to 1.

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    2. First, an must be positive integer.
      by putting n=1,n=2 ,n=3 into an = (n+1)(2n+1)/2 - n -(2n+1) = (2n^2 - 3n -1)/2,
      we may get a negative number or fraction.

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    3. You are right.
      a1=0=1*0
      a2=2=2*1
      a3=6=3*2
      a4=12=4*3
      :

      So actually, an = n(n-1)

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    4. You are right.
      a1=0=1*0
      a2=2=2*1
      a3=6=3*2
      a4=12=4*3
      :

      So actually, an = n(n-1)

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