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\title{Math 214 -- Foundations of Mathematics\\
Homework 4\\
{\large{\bf Due Sept 28, 2012}}}
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\author{Your name}


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Solve the following problems.  Show all your work.  Every problem worths 4 points. 

\begin{enumerate}
\item Prove that $\sum_{k=1}^{n} \frac{1}{(k+2)(k+3)}=\frac{n}{3n+9}$ for every positive integer $n$. \\

\item Prove that $(2012)!>2^{2012}$.  (Hint: prove a general statement in terms of $n$.)\\

\item For each of the following sets, determine whether it is well-ordered and show your reasons. \\
(a) $S=\{n\in \N: n \text{ is even}\}$.\\
(b) $T=\{n\in \Z: n \text{ is odd}\}$. 

\item A sequence $\{a_n\}$ is defined recursively by $a_1=1, a_2=4, a_3=9$, and $$a_n=a_{n-1}-a_{n-2}+a_{n-3}+2(2n-3)$$ for $n\ge 4$.  Conjecture a formula for $a_n$ and prove that your conjecture is correct. \\


\item Use Strong Principle of Mathematical Induction to prove that for each integer $n\ge 10$,  there are nonnegative integers $a$ and $b$ such that $n=2a+7b$.  \\

\item Prove that $7 |  (3^{2n}-2^n)$ for every nonnegative integer $n$. \\
\end{enumerate}

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