Question

A sequence of numbers is defined by $D_0 = 0,$ $D_1 = 0,$ $D_2 = 1,$ and $D_n = D_{n-1} + D_{n-3}$ for $n \ge 3.$ What are the parities (evenness or oddness) of the triple of numbers $(D_{2021}, D_{2022}, D_{2023}),$ where $E$ denotes even and $O$ denotes odd?

$(O, E, O)$

$(E, E, O)$

$(E, O, E)$

$(O, O, E)$

$(O, O, O)$

Answer: C

Concepts:parity recursion pattern recognition

Difficulty rating: 1600

Solution:

Working modulo $2,$ the terms $D_0, D_1, D_2, \ldots$ have parities $E,$ $E,$ $O,$ $O,$ $O,$ $E,$ $O,$ $E,$ $E,$ $O,$ $O,$ $O,$ $E,$ $O, \ldots$ which repeat with period $7$ starting from $D_0$ (indeed $D_7, D_8, D_9$ have the same parities $E, E, O$ as $D_0, D_1, D_2$ ).

Since $2021 \equiv 5,$ $2022 \equiv 6,$ and $2023 \equiv 0 \pmod 7,$ the parities match those of $D_5, D_6, D_0,$ namely $E, O, E.$

Thus, the correct answer is C.

2021 AMC 12A Spring Problem 8

Answer: C

Difficulty rating: 1600

Solution:

Problem 8 in Other Years