Problem:
In the eight-term sequence A,B,C,D,E,F,G,H, the value of C is 5 and the sum of any three consecutive terms is 30. What is A+H?
Answer Choices:
A. 17
B. 18
C. 25
D. 26
E. 43
Solution:
Note that for any four consecutive terms, the first and last terms must be equal. For example, consider B,C,D, and E; because
B+C+D=30=C+D+E,
we must have B=E. Hence A=D=G, and C=F=5. The required sum A+H=G+(30−G−F)=30−5=25.
OR
Note that
A+C+H=(A+B+C)−(B+C+D)+(C+D+E)−(E+F+G)+(F+G+H)=3⋅30−2⋅30=30.
Hence A+H=30−C=(C)25.
The problems on this page are the property of the MAA's American Mathematics Competitions