Problem:
Consider the sequence of numbers: For , the th term of the sequence is the units digit of the sum of the two previous terms. Let denote the sum of the first terms of this sequence. The smallest value of for which is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Writing out more terms of the sequence yields
The sequence repeats itself, starting with the 13 th term. Since for all positive integers . The largest for which is
and . To have , we need to add enough additional terms for their sum to exceed 40 . This can be done by adding the next 7 terms of the sequence, since their sum is 42 . Thus, the smallest value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions