2013 AMC 10B Problems/Problem 21
Problem
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of N?
Solution
Let the first two terms of the first sequence be and
and the first two of the second sequence be
and
. Computing the seventh term, we see that
. Note that this means that
and
must have the same value modulo 8. To minimize, let one of them be 0; WLOG assume that
. Thus, the smallest possible value of
is
; since the sequences are nondecreasing
. To minimize, let
. Thus,
.
I don't know why the person got the answer above, but the answer seems to be 55. if and
are 11 and 0, respectively, and
and
are 3 and 5, respectively, the answer would be
.