Problem:
A sequence of numbers is defined by D0=0,D1=0,D2=1 and Dn=Dn−1+Dn−3 for n≥ 3. What are the parities (evenness or oddness) of the triple of numbers (D2021,D2022,D2023), where E denotes even and O denotes odd?
Answer Choices:
A. (O,E,O)
B. (E,E,O)
C. (E,O,E)
D. (O,O,E)
E. (O,O,O)
Solution:
The parities of the first 10 terms of the sequence (Dn) are as follows:
nParity0E1E2O3O4O5E6O7E8E9O
The terms D7,D8, and D9 have the same parities as D0,D1,D2, respectively, so the sequence is periodic with period 7. The remainder when 2021 is divided by 7 is 5 , so the parities of (D2021,D2022,D2023) are the same as the parities of D5,D6,D7, namely (E,O,E).
The problems on this page are the property of the MAA's American Mathematics Competitions