Let P_0(x)=x^3+313x^2-77x-8. For integers n≥1, define P_n(x)=P_n-1(x-n). What is the coefficient of x in P_20(x)?

1993 AIME Problem 5

Attempt Problem 5 of the 1993 AIME below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1993 AIME solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Let

P_0(x)=x^3+313x^2-77x-8.

For integers

n\geq1,

define

P_n(x)=P_{n-1}(x-n).

What is the coefficient of

x

P_{20}(x)?

Answer

Collapse the repeated shifts to write

P_{20}

directly in terms of

P_0

Only collect the linear terms after substituting

x-210

The accumulated shift is

1+2+\cdots+20=210,

P_{20}(x)=P_0(x-210).

The coefficient of

x

is therefore

\begin{aligned}&3(210)^2-2(313)(210)-77\\&\quad=132300-131460-77\\&\quad=763.\end{aligned}