How do you find the explicit formula for the following sequence 1/2,3/4,5/8,7/16 ...?

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Answer 1 · 2016-05-30T17:07:45

#a_n = (2n-1)/2^n#

The numerators form an arithmetic sequence:

#1, 3, 5, 7#

with common difference #2#.

So a formula for the numerator could be written:

#p_n = 1+2(n-1) = 2n-1#

The denominators form a geometric sequence:

#2, 4, 8, 16#

with common ratio #2#.

So a formula for the denominator could be written:

#q_n = 2*2^(n-1) = 2^n#

Thus a formula for a general term of our example sequence can be written:

#a_n = p_n/q_n = (2n-1)/2^n#