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\begin{document}

\begin{flushright}
Your Name Here\\
Math 220\\
HW \# 6\\
\today
\end{flushright}

%Please write your solutions in the designated solution area!

\begin{ex}
Prove that for all integers $n\geq1$,
$$1^2+2^2+\cdots+n^2= \dfrac{n(n+1)(2n+1)}{6}.$$
\end{ex}

\begin{proof}[Solution]

\end{proof}

\begin{ex}
Prove that for all integers $n\geq 1$,
$$\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot 4}+\cdots+\dfrac{1}{(n+1)(n+2)} = \dfrac{n}{2n+4}.$$
(Please do this using induction. While it would be valid to just take the result from class and subtract $\frac{1}{2}$, do not do this.)
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove that for integers $n\geq1$,
$$1\cdot 1!+2\cdot 2!+\cdots+n\cdot n! = (n+1)!-1.$$
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove that for all integers $n\geq0$, $3|(2^{2n}-1)$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Observe the pattern
\begin{eqnarray*}
\left(1-\dfrac{1}{2}\right) &=& \dfrac{1}{2}\\
\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right) &=& \dfrac{1}{3}\\
\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right) &=& \dfrac{1}{4}
\end{eqnarray*}
Conjecture a general formula for integers $n\geq2$, and use induction to prove that formula.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Prove that for all integers $n\geq10$, $2^n> n^3$.
\end{ex}

\begin{proof}

\end{proof}

\begin{ex}
Consider the open sentence $P(n): 9+13+\cdots+(4n+5)=\dfrac{4n^2+14n+1}{2}$, where $n\in\mathbb{N}$.
\begin{enumerate}[$(a)$]
\item Verify the implication $P(k)\Rightarrow P(k+1)$ for an arbitrary positive integer $k$.
\item Is $\forall n\in\mathbb{N}, P(n)$ a true statement?
\end{enumerate}
\end{ex}

\begin{proof}

\end{proof}

\end{document}