3. Montasser creates a sequence of numbers. The first two numbers are 6 and 15. He generates each subsequent number in the sequence by dividing the last number by the one before it and then multiplying the result by 2. Thus, the third number in the sequence is $\frac{15}{6} \cdot 2=5$ and the fourth number is $\frac{5}{15} \cdot 2=\frac{2}{3}$.

What is the hundredth number in the sequence?
A) 15
B) 5
C) $\frac{2}{3}$
D) $\frac{4}{15}$
E) $\frac{4}{5}$
A3. C) $\frac{2}{3}$ We calculate the following numbers in this sequence:

- 6 ;
- 15 ;
- $\frac{15}{6} \cdot 2=5$
- $\frac{5}{15} \cdot 2=\frac{2}{3}$;
- $\frac{2}{3} \div 5 \cdot 2=\frac{2}{3} \cdot \frac{1}{5} \cdot 2=\frac{4}{15}$;
- $\frac{4}{15} \div \frac{2}{3} \cdot 2=\frac{4}{15} \cdot \frac{3}{2} \cdot 2=\frac{4}{5}$;
- $\frac{4}{5} \div \frac{4}{15} \cdot 2=\frac{4}{5} \cdot \frac{15}{4} \cdot 2=6$;
- $6 \div \frac{4}{5} \cdot 2=6 \cdot \frac{5}{4} \cdot 2=15$;
- ...

We see that the sequence is periodic: the same six numbers repeat themselves. The first number is 6, the seventh number is 6, ..., the 97th number is 6. The hundredth number is therefore $\frac{2}{3}$.