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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 solution | Comment 1 of 2
It hardly matters what the terms of S(n) are.  Since they are powers of 3 and we keep taking logarithms, it's just the exponents that matter.  Let 3^(E(n)) = S(n). If the terms of S(n) are geometric, than the exponents E(n) are arithmetic.

The first given is that the first 8 exponents have sum 308
The sum is 28a+8b
Possibilities: (a,b,E(7))
If we sum terms 0 through 7 of S(n) the answer isn't much larger than the last term, so its base 3 logarithm isn't much bigger than the last exponent.  So the (5,21,56) is our sequence.

The 14th term of E(n) is therefore 91 = log3S(14)

  Posted by Jer on 2016-02-22 11:03:46
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