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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Given the sequence:

equation

what is the difference between the 21st and the 26th terms?

The sequence has alternating definitions meaning that it is practically made up of two sequences whose terms are alternating. The odd terms form the sequence:

equation

while the even terms form the sequence:

equation

Both sequences are arithmetic.

The main difficulty is establishing the rank of the needed term in the respective half-sequence. We need to account properly for the rank:

TermRank in
alternating
sequence
Rank in
Sequence #1
Rank in
Sequence #2
111=2/2n/a
02n/a1=2/2
232=4/2n/a
-14n/a2=4/2
353=6/2n/a
-26n/a3=6/2
474=8/2n/a

A rank of 21st corresponds to Sequence #1. Notice that the rank r1 of a term in Sequence #1 can be calculated from the rank r of the alternate sequence using:

equation

and the term is actually equal to the rank:

equation

A rank of 26st corresponds to Sequence #2. Notice that the rank r2 of a term in Sequence #2 can be calculated from the rank r of the alternate sequence using:

equation

and that we can get the term by subtracting the rank from 1:

equation

The difference is:

equation