The seven colors keep repeating in the same order.
Color: red orange yellow green blue indigo violet
Bead
Number: 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 etc.
The violet beads are in positions 7,14,21...,70,...,that is multiples of 7.
If 101 beads were a multiple of 7, the 101st bead would be violet, but when 101 is divided by 7, the quotient would be 14 and a remainder of 3.
Since 14*7=98, the 98th bead completes the 14th cycle, and hence is violet.
The 99 starts the next cycle; is is red.
The 100th bead is Orange, and the 101st bead is yellow.
Answer: Yellow