1986 AIME Problems/Problem 11
Problem
The polynomial may be written in the form
, where
and the
's are constants. Find the value of
.
Contents
Solution
Solution 1
Using the geometric series formula, . Since
, this becomes
. We want
, which is the coefficient of the
term in
(because the
in the denominator reduces the degrees in the numerator by
). By the Binomial Theorem, this is
.
Solution 2
Again, notice . So
We want the coefficient of the term of each power of each binomial, which by the binomial theorem is
. The Hockey Stick Identity tells us that this quantity is equal to
.
Solution 3
Again, notice . Substituting
for
in
gives:
From binomial theorem, the coefficient of the
term is
. This is actually the sum of the first 16 triangular numbers, which evaluates to
.
Solution 4(calculus)
Let and
.
Then, since ,
by the power rule.
Similarly,
Now, notice that if , then
, so
, and
.
Now, we can use the hockey stick theorem to see that
Thus,
See also
1986 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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