, 2011

Jackson gave the following rule to create sequences: "If $x$ is a term in your sequence, then the next term in your sequence is $\dfrac{1}{1-x}$." For example, Mary starts her sequence with the number 3. The second term of her sequence is $\dfrac{1}{1-3}=\dfrac{1}{-2}=-\dfrac{1}{2}$. Her sequence is now $3, -\dfrac{1}{2}$. The third term of her sequence is $\dfrac{1}{1-(-\frac{1}{2})}=\dfrac{1}{\frac{3}{2}}=\dfrac{2}{3}$. Her sequence is now $3, -\dfrac{1}{2}, \dfrac{2}{3}$. The fourth term of her sequence is $\dfrac{1}{1-\frac{2}{3}}=\dfrac{1}{\frac{1}{3}}=3$. Her sequence is now $3, -\dfrac{1}{2}, \dfrac{2}{3}, 3$. Fabien starts his sequence with the number 2, and continues using Jackson's rule until the sequence has 2011 terms. (a) What is the second term of his sequence? (b) What is the fifth term of his sequence? (c) How many of the 2011 terms in Fabien's sequence are equal to 2? Explain. (d) Determine the sum of all of the terms in his sequence.