Wednesday, June 25, 2008

`1/8, 1/4, 1/2, 1...` Determine whether the sequence is geometric. If so, find the common ratio.

You need to verify if the sequence is a geometric sequence, hence, you need to use the following property, such that:


`a_n = sqrt(a_(n-1)*a_(n+1))`


Replacing `1/4` for `a_n, 1/8` for `a_(n-1)` and `1/2` for `a_(n+1)` yields:


`1/4 = sqrt(1/8*1/2) => 1/4 = sqrt(1/16) => 1/4 = 1/4`


It also may be verified if `1/2 = sqrt(1/4*1) => 1/2=1/2.`


Since the given sequence is geometric, you may find the ratio such that:


`q = (a_(n+1))/(a_n) => q = 1/(1/2) => q = 2`


Hence, the given sequence is geometric and its ratio is q = 2.

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