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)
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
E(n)=an+b
The sum is 28a+8b
Possibilities: (a,b,E(7))
(1,35,42)
(3,28,49)
(5,21,56)
(7,14,63)
(9,7,70)
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 = log_{3}S(14)

Posted by Jer
on 20160222 11:03:46 