Using only digits 1, 2, 3, 4, and 5, a sequence is created as follows: one 1, two 2's, three 3's, four 4's, five 5 's, six 1's, seven 2's, and so on.

The sequence appears as: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,1,1,1,1,1,1,2,2, \ldots$.

The 100th digit in the sequence is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Using only digits $1,2,3,4$, and 5 , a sequence is created as follows: one 1 , two 2 's, three 3 's, four 4's, five 5 's, six 1's, seven 2's, and so on.

The sequence appears as: $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,1,1,1,1,1,1,2,2, \ldots$.

The 100th digit in the sequence is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

## Solution

The total number of digits in $n$ groups of the sequence is given by $1+2+3+\ldots+n$. In order to determine the group containing the 100th digit in the sequence, we must find the positive integer $n$ such that $1+2+3+\ldots+(n-1)100$. By examining a few of these sums we find that $1+2+3+\ldots+13=91$ and $1+2+3+\ldots+13+14=105$. Thus the 100th digit in the sequence is in the 14th group. The 100th digit is a 4.

ANSWER: (D)