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Harmonic Sequence Settlement (Posted on 2014-05-22) Difficulty: 3 of 5
Determine the total number of ways in which 363 can be written as a sum of three distinct positive integers which are in harmonic sequence.

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution solution with computer code and derivation | Comment 2 of 3 |
1/c - 1/b = 1/b - 1/a

ab - ac = ac - bc
ab + bc = 2ac
b(a + c) = 2ac

b = 2ac/(a+c)

Let a be the smallest and c the largest:


DefDbl A-Z
Dim crlf$

Private Sub Form_Load()
 Text1.Text = ""
 crlf$ = Chr(13) + Chr(10)
 Form1.Visible = True
 DoEvents
 n = 363
 For a = 1 To n / 3
  For c = a + 1 To n - a
    b = 2 * a * c / (a + c)
    If b = Int(b) Then
      If a + b + c = n Then
        Text1.Text = Text1.Text & Str(a) & Str(b) & Str(c) & crlf
      End If
    End If
  Next
 Next a
End Sub



finds 

 13 25 325
 55 88 220
 66 99 198
 108 120 135
 

  Posted by Charlie on 2014-05-22 16:34:42
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