jmc

Question

Consider the sequence of numbers: 4,7,1,8,9,7,6, For n>2, the n-th term of the sequence is the units digit of the sum of the two previous terms. Let S_n denote the sum of the first n terms of this sequence. Find the smallest value of n for which S_n>10,000.

Accepted Answer

Writing out further terms of the sequence, we get

4, 7, 1, 8, 9, 7, 6, 3, 9, 2, 1, 3, 4, 7, \dots .

Since each term depends only on the previous two terms, the sequence becomes periodic at this point, with period 12.

Since

S_{12} = 60,

S_{12 k} = 60 k

for all positive integers

k .

Taking

k = 166,

we get

S_{1992} = 60 \cdot 166 = 9960.

Then

S_{1998} = 9996

and

S_{1999} = 10002,

so the smallest such

n

is

1999 .