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{\bf CSci 243 Homework 4}
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**My name**
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\begin{enumerate}

\item (12 points, 3 points each)
For each of these partial sequences of integers, determine the next term of 
the sequence, and then provide a general formula or rule to generates terms of the
sequence.
\begin{enumerate}
\item $3, 6, 11, 18, 27, 38, 51, 66, 83, 102, \ldots$
\item $1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, \ldots$
\item $0, 2, 8, 26, 80, 242, 728, 2186, 6560, 19682, \ldots$
\item $2, 4, 16, 256, 65536,4294967296,  \ldots$
\end{enumerate}

\item(10 points, 2 points each) Compute each of these sums.
\begin{enumerate}
\item $\sum_{i=1}^3\sum_{j=1}^2(i-j)$
\item $\sum_{i=0}^3\sum_{j=0}^2(3i+2j)$
\item $\sum_{i=1}^3\sum_{j=0}^2j$
\item $\sum_{i=0}^2\sum_{j=0}^3i^2j^3$
\item $\sum_{i=99}^{200} (i+1)i(i-1)$
\end{enumerate}

\item (7 points) Find the summation formulae of $\sum_{k=0}^n (a + kd)$, i.e., the close-form expression for the summation of general arithmetic progression, by telescoping. 

\item (8 points) Find the closed-form expression for $\sum_{k=2}^n\frac{1}{k^2-1}$ by telescoping.


\item (8 points) Find the closed-form expressions for the following.  Show your work.
\begin{enumerate}
\item (2 points) $\sum_{i=0}^n 3^{i+1}-3^i$
\item (6 points) $\sum_{i=0}^{2n} (-2)^i$  (hint: split series in two parts)
\end{enumerate}

\item (\textbf{Bonus Question} 1 points\footnote{The points for bonus questions are added directly to the final grade.}) Let $a_n$, where $n \geq 0$ be the $n_{th}$ term for the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, ... -- constructed by including the integer $k$ exactly $k$ times. Find the closed-form for $a_n$. You don't need to explain the answer. 
 
\end{enumerate}

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