The polynomial 1-x+x^2-x^3+… +x^16-x^17 may be written in the form &a_0+a_1y+a_2y^2+… & +a_16y^16+a_17y^17, where y=x+1 and the a_i ’s are constants. Find a_2.

1986 AIME Problem 11

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11.

The polynomial

1-x+x^2-x^3+\cdots

{}+x^{16}-x^{17}

may be written in the form

\begin{aligned} &a_0+a_1y+a_2y^2+\cdots\\ &\qquad{}+a_{16}y^{16}+a_{17}y^{17}, \end{aligned}

where

y=x+1

and the

a_i

’s are constants. Find

a_2.

Answer

Substitute

x=y-1

before expanding

Find only the coefficient of

y^2

in each power and use the hockey-stick identity

The polynomial is

\sum_{k=0}^{17}(-x)^k.

Since

x=y-1,

this becomes

\sum_{k=0}^{17}(1-y)^k.

For

k\geq2,

the coefficient of

y^2

(1-y)^k

\binom{k}{2}.

Hence

a_2=\sum_{k=2}^{17}\binom{k}{2} =\binom{18}{3}=816.