2016 AIME II Problems/Problem 9
The sequences of positive integers and
are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let
. There is an integer
such that
and
. Find
.
Solution
Since all the terms of the sequences are integers, and 100 isn't very big, we should just try out the possibilities for . When we get to
and
, we have
and
, which works, therefore, the answer is
.
Solution by Shaddoll
Solution 2
Using the same reasoning ( isn't very big), we can guess which terms will work. The first case is
, so the second and fourth terms of
are
and
. We let
be the common ratio of the geometric sequence and write the arithmetic relationships in terms of
.
The common difference is , and so we can equate:
. Moving all the terms to one side and the constants to the other yields
, or
. Simply listing out the factors of
shows that the only factor
less than a square that works is
. Thus
and we solve to get
from there.
Solution by rocketscience