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Initial Term Illation (Posted on 2016-02-25) Difficulty: 3 of 5
The sequence {S(n)} is such that:
S(0) is a positive integer, and:
S(n) = 5*S(n-1)+4 for n ≥ 1

The task is to choose S(0) such that 2016 divides S(54).

Does there exist an infinite number of values that can be assigned to S(0)?
Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 2 of 2 |
DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For s0 = 1 To 99999
   DoEvents
   s = s0
   For i = 1 To 54
     s = (s * 5 + 4) Mod 2016
   Next
   If s = 0 Then
     Text1.Text = Text1.Text & s0 & crlf
   End If
 Next


 Text1.Text = Text1.Text & crlf & " done"
  
End Sub

finds

504
2520
4536
6552
8568
10584
12600
14616
16632
18648
20664
22680
24696
26712
28728
30744
32760
34776
36792
38808
40824
42840
44856
46872
48888
50904
52920
54936
56952
58968
60984
63000
65016
67032
69048
71064
73080
75096
77112
79128
81144
83160
85176
87192
89208
91224
93240
95256
97272
99288

so any 504 + 2016k works, where k=0, 1, ...

Starting with s(0)=504, s(54) and s(54)/2016 are respectively

 28033131371785202645696699619293212890624
 
 and 
 
 13905323101083929883778124811157347664.0 
 
for the next possible value of s(0), 2520, these two numbers are

 139943612254000981920398771762847900390624 
   and
 69416474332341756904959708215698363289.0 
 
 based upon

    5   open "inttrmil.txt" for output as #2
   10   S=504
   20   for I=1 to 54
   30       S=S*5+4
   40   next
   50   print #2,S
   60   print #2,S/2016
   70   close #2


  Posted by Charlie on 2016-02-25 16:03:19
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