2015 AIME II Problems/Problem 13
Contents
Problem
Define the sequence by
, where
represents radian measure. Find the index of the 100th term for which
.
Solution 1
If ,
. Then if
satisfies
,
, and
Since
is positive, it does not affect the sign of
. Let
. Now since
and
,
is negative if and only if
, or when
. Since
is irrational, there is always only one integer in the range, so there are values of
such that
at
. Then the hundredth such value will be when
and
.
Solution 2
Notice that is the imaginary part of
, by Euler's formula. Using the geometric series formula, we find that this sum is equal to
We only need to look at the imaginary part, which is
Since
,
, so the denominator is positive. Thus, in order for the whole fraction to be negative, we must have
. This only holds when
is between
and
for integer
[continuity proof here], and since this has exactly one integer solution for every such interval, the
th such
is
.
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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