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Geometric and Three Power Travail (Posted on 2016-02-22) Difficulty: 3 of 5
S(0), S(1), S(2) , ..... are terms of a geometric sequence in strictly ascending orders of magnitude.
All the terms of this sequence are nonnegative integer powers of 3, like:
30, 31, .... etc

Given that:
Σn=0 to 7 (log3S(n)) = 308, and:

56 ≤ (log3 n=0 to 7 S(n)))≤ 57

Find log3S(14)

No Solution Yet Submitted by K Sengupta    
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Solution computer assisted solution Comment 2 of 2 |
Arithmetic series with 8 members totaling 308:

start with 0, difference 11: resulting sum of powers has base-3 log = 77.0000051383416
start with 7, difference 9: resulting sum of powers has base-3 log = 70.0000462461185
start with 14, difference 7: resulting sum of powers has base-3 log = 63.0004162996764
start with 21, difference 5: resulting sum of powers has base-3 log = 56.0037535691403
start with 28, difference 3: resulting sum of powers has base-3 log = 49.0343527269525
start with 35, difference 1: resulting sum of powers has base-3 log = 42.3689315010237

based on

 For st = 0 To 40
  For diff = 1 To 15
    tot = 0
    For i = 0 To 7
      tot = tot + st + i * diff
    Next
    If tot = 308 Then
      term = 3 ^ st: t2 = term
      For i = 1 To 7
        term = term * 3 ^ diff
        t2 = t2 + term
      Next
      Text1.Text = Text1.Text & st & Str(diff) & Str(Log(t2) / Log(3)) & crlf
    End If
  Next
 Next

Therefore S(0)=3^21, S(2)=3^26, ..., S(14) = 3^(21+5*14) = 3^91, whose base-3 log is 91

  Posted by Charlie on 2016-02-22 12:17:23
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