For a sequence u_1, u_2, …, define Δ^1(u_n)=u_n+1-u_n and, for all integers k>1, Δ^k(u_n)=Δ^1(Δ^k-1(u_n)). If u_n=n^3+n, then Δ^k(u_n)=0 for all n

1976 AMC 12 Problem 25

Attempt Problem 25 of the 1976 AMC 12 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 1976 AMC 12 solutions, or check the answer key.

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25.

For a sequence

u_1,

u_2,

\ldots,

define

\Delta^1(u_n)=u_{n+1}-u_n

and, for all integers

k\gt1,

\Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n)).

u_n=n^3+n,

then

\Delta^k(u_n)=0

for all

n

k=1

k=2,

but not if

k=1

k=3,

but not if

k=2

k=4,

but not if

k=3

for no value of

k

Each forward difference lowers the degree of a polynomial sequence by one

Compute through the constant third difference to see when zero first appears

Directly,

\begin{aligned} \Delta u_n&=3n^2+3n+2,\\ \Delta^2u_n&=6n+6,\\ \Delta^3u_n&=6,\\ \Delta^4u_n&=0. \end{aligned}

Thus the fourth differences vanish for all

n,

but the third differences do not.

Therefore, the correct answer is D.