1986 AIME Problems/Problem 11

Revision as of 19:21, 12 September 2007 by Azjps (talk | contribs) (cat, link, minor)

Problem

The polynomial $1-x+x^2-x^3+\cdots+x^{16}-x^{17}$ may be written in the form $a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}$, where $\displaystyle y=x+1$ and the $\displaystyle a_i$'s are constants. Find the value of $\displaystyle a_2$.

Solution

Since $(1+x)(1-x+x^2-x^3+\cdots +x^{16}-x^{17})=1-x^{18}$, we have

$y(a_0+a_1y+a_2y^2+\cdots +a_{17}y^{17})=1-(y-1)^{18}$

So $a_2$ is the $y^3$ coefficient, which, by the Binomial Theorem, is $\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=3\cdot 17\cdot 16=816$

See also

1986 AIME (ProblemsAnswer KeyResources)
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
Invalid username
Login to AoPS