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\title[Math 250 Handout 4 - Induction warmup]{Math 250 Handout 4 - Induction warmup}\maketitle

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1. We want to prove, by induction, that, for every positive integer $n$, 
\[
1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}.
\]
\begin{itemize}
\item [a)] What is the open statement ``$P(n)"$?\\

P(n) = 
\vspace{1cm}
\item [b)] What is the statement ``$P(1)"$? Why is $P(1)$ true?\\

P(1) = 
\vspace{1cm}
\item [c)] What is the inductive step? Write out your assumption, your desired conclusion, and the inductive step (i.e., the proof that $P(n-1) \Rightarrow P(n)$).

 Assume that \\\vspace{0.5cm}

 We want to show that\\\vspace{0.5cm}

 (Inductive step)\\\vspace{0.5cm}

\end{itemize}

2. Let $a_n$ be a sequence such that $a_1 = 1$ and $a_n = na_{n-1}$. We want to prove, by induction, that, for every positive integer $n$, 
\[
a_n = n! = n(n-1)(n-2)\cdots 2\cdot 1.
\]
\begin{itemize}
\item [a)] What is the open statement ``$P(n)"$?\\

P(n) = 
\vspace{0.5cm}
\item [b)] What is the statement ``$P(1)"$? Why is $P(1)$ true?\\

P(1) = 
  \newpage
\vspace{0.5cm}
\item [c)] What is the inductive step? Write out your assumption, your desired conclusion, and the inductive step (i.e., the proof that $P(n-1) \Rightarrow P(n)$).

 Assume that \\\vspace{0.5cm}

 We want to show that\\\vspace{0.5cm}

 (Inductive step)\\\vspace{0.5cm}

\end{itemize}

3. We want to prove, by induction, that, for every positive integer $n$, 
\[
1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{n^2(n+1)^2}{4}.
\]
\begin{itemize}
\item [a)] What is the open statement ``$P(n)"$?\\

P(n) = 
\vspace{1cm}
\item [b)] What is the statement ``$P(1)"$? Why is $P(1)$ true?\\

P(1) = 
\vspace{1cm}
\item [c)] What is the inductive step? Write out your assumption, your desired conclusion, and the inductive step (i.e., the proof that $P(n-1) \Rightarrow P(n)$).

 Assume that \\\vspace{0.5cm}

 We want to show that\\\vspace{0.5cm}

 (Inductive step)\\\vspace{0.5cm}

\end{itemize}

4. Prove, by induction, that $2^{n+1} \geq n^2$ for every integer $n$. (For this problem, you will have to first check $P(1)$ and $P(2)$.)

\end{document}
