Just using the finite first 20 and first 16 sequences:
The first terms and the last terms of each of these finite sequences (20 members and 16 members) are equal. That would be impossible if both were positive. So A is negative.
The program tests the possibilities:
DefDbl AZ
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For a = 1 To 1000 Step 1
For d1 = 1 To 2000
fin = a + 19 * d1
tot = 20 * (a + fin) / 2
d2 = (fin  a) / 15
If d2 = Int(d2) Then
If tot = 16 * (a + fin) / 2 Then
Text1.Text = Text1.Text & a & Str(fin) & " " & d1 & Str(d2) & crlf
End If
End If
Next
Next a
Text1.Text = Text1.Text & crlf & " done"
End Sub
The two totals are of course equal at zero, and the minimum absolute value of A is 285, when the sequences go from 285 to 285 with differences of 30 and 38 respectively.
285 285 30 38
570 570 60 76
855 855 90 114
On hindsight of course, 285 if the LCM of 19 and 15, the number of differences between members of the sequences.

Posted by Charlie
on 20160504 13:13:53 