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Seventh Term Settlement (Posted on 2016-07-22) Difficulty: 3 of 5
Each of A(1), A(2),..., A(7) is a positive integer such that:

(i) A(6) = 144, and

(ii) A(n+3) = A(n+2)(A(n+1)+A(n)), for n = 1,2,3,4

Find A(7)

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 2 of 10 |
First let's make (ii) more manageable by writing the recursion in more usual terms:

A(n) = A(n-1)*(A(n-2) + A(n-3)) for n = 4, 5, 6 and 7

DefDbl A-Z
Dim crlf$, a(7)


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For a1 = 1 To 36
   a(1) = a1
 For a2 = 1 To 36
   a(2) = a2
 For a3 = 1 To 36
   DoEvents
   a(3) = a3
   For i = 4 To 7
     a(i) = a(i - 1) * (a(i - 2) + a(i - 3))
   Next i
   If a(6) = 144 Then
     For i = 1 To 7
       Text1.Text = Text1.Text & Str(a(i))
     Next i
     Text1.Text = Text1.Text & crlf
   End If
 Next
 Next
 Next
 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

resulting in:

 2 1 2 6 18 144 3456
 7 1 1 8 16 144 3456

The results show that A(6) is 144 only if A(1), A(2) and A(3) are either 2, 1 and 2; or 7, 1 and 1, respectively.  Either way A(7) is 3456.

  Posted by Charlie on 2016-07-22 10:34:02
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