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:
3^{0}, 3^{1}, .... etc
Given that:
Σ_{}_{n=0 to 7} (log_{3}S(n)) = 308, and:
56 ≤ (log_{3}(Σ_{ n=0 to 7} S(n)))≤ 57
Find log_{3}S(14)
Arithmetic series with 8 members totaling 308:
start with 0, difference 11: resulting sum of powers has base3 log = 77.0000051383416
start with 7, difference 9: resulting sum of powers has base3 log = 70.0000462461185
start with 14, difference 7: resulting sum of powers has base3 log = 63.0004162996764
start with 21, difference 5: resulting sum of powers has base3 log = 56.0037535691403
start with 28, difference 3: resulting sum of powers has base3 log = 49.0343527269525
start with 35, difference 1: resulting sum of powers has base3 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 base3 log is 91.

Posted by Charlie
on 20160222 12:17:23 