1986 AIME Problems/Problem 11

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 $\frac{18\cdot 17\cdot 16}{3\cdot 2\cdot 1}=3\cdot 17\cdot 16=816$

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
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1986 AIME Problems/Problem 11

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