SOLUTION: log(2)8, log(4)8, log(8)8, log(16)8, log(32)8, log(64)8, log(128)8 find an expression of the nth term of the sequence. write expression in p/q form. justify using technology.

Algebra ->  Algebra  -> Logarithm Solvers, Trainers and Word Problems -> SOLUTION: log(2)8, log(4)8, log(8)8, log(16)8, log(32)8, log(64)8, log(128)8 find an expression of the nth term of the sequence. write expression in p/q form. justify using technology.      Log On

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Question 161686: log(2)8, log(4)8, log(8)8, log(16)8, log(32)8, log(64)8, log(128)8
find an expression of the nth term of the sequence. write expression in p/q form. justify using technology.
Answer by Earlsdon(6103) About Me  (Show Source):
You can put this solution on YOUR website!
Find the nth term (a%5Bn%5D):
Log%5B2%5D%288%29,Log%5B4%5D%288%29,Log%5B8%5D%288%29,Log%5B16%5D%288%29,Log%5B32%5D%288%29,Log%5B64%5D%288%29,Log%5B128%5D%288%29,...
You can see that the base of the logarithms is always 2 to some power, and the power of 2 goes up like this, starting with the first term a%5B1%5D:2%5E1+=+2,2%5E2+=+4,2%5E3+=+8,2%5E4+=+16,2%5E5+=+32,2%5E6+=+64,2%5E7+=+128,..., so for the nth term a%5Bn%5D the base would be:2%5En
Now you can write the nth term as:
a%5Bn%5D+=+Log%5B2%5En%5D%288%29 Now if you rewrite this in exponential form, you'll get:
%282%5En%29%5Ea%5Bn%5D+=+8 Substitute 8+=+2%5E3 to get:
2%5E%28%28n%2Aa%5Bn%5D%29%29+=+2%5E%28%283%29%29 Now since the bases (2) are equal, the exponents must be equal, so...
n%2Aa%5Bn%5D+=+3 Divide both sides by n to get:
highlight%28a%5Bn%5D+=+3%2Fn%29 This is the expression for the nth term expressed in p%2Fq form.