perplexus.info :: Sequences : Initial Term Illation

Initial Term Illation (Posted on 2016-02-25)

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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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

If s = 0 Then

Text1.Text = Text1.Text & s0 & crlf

End If

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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