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\ihead{Math 221 Winter 2025 (Darij Grinberg): homework set 2}
\ohead{page \thepage}
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\begin{document}

\begin{center}
\textbf{Math 221 Winter 2025 (Darij Grinberg): homework set 2}

due date: Sunday 2025-01-23 at 11:59PM on gradescope (
\url{https://www.gradescope.com/courses/930212} ).

Please solve only \textbf{3 of the 6 exercises}.
\end{center}

Recall that $\mathbb{N}=\left\{  0,1,2,\ldots\right\}  $.

\begin{exercise}
\textbf{(a)} Prove that
\[
\prod_{i=2}^{n}\left(  1-\dfrac{1}{i^{2}}\right)  =\dfrac{n+1}{2n}%
\]
for each positive integer $n$. \medskip

\textbf{(b)} Find and prove a closed-form expression (i.e., no $\prod$ or
$\sum$ signs) for%
\[
\prod_{i=2}^{n}\left(  1-\dfrac{1}{i}\right)  .
\]

\end{exercise}

\begin{exercise}
Prove that%
\[
\prod_{i=1}^{n}\left(  i!\cdot i^{i}\right)  =n!^{n+1}%
\ \ \ \ \ \ \ \ \ \ \text{for each }n\in\mathbb{N}.
\]

\end{exercise}

\bigskip

The \emph{floor} $\left\lfloor x\right\rfloor $ of a real number $x$ means the
largest integer that is smaller or equal to $x$. For instance, $\left\lfloor
6.2\right\rfloor =6$ and $\left\lfloor 7.7\right\rfloor =7$ and $\left\lfloor
8\right\rfloor =8$. (In other words, $\left\lfloor x\right\rfloor $ is what
you get if you round $x$ down. Beware: $\left\lfloor -1.3\right\rfloor $ is
$-2$, not $-1$.)

\begin{exercise}
Let $n\in\mathbb{N}$. Prove that%
\[
\sum_{k=1}^{n}\left\lfloor \dfrac{k}{2}\right\rfloor =\left\lfloor \dfrac
{n}{2}\right\rfloor \cdot\left\lfloor \dfrac{n+1}{2}\right\rfloor .
\]

\end{exercise}

(In this exercise, you can freely use basic properties of even and odd numbers
-- such as Proposition 3.3.8 in the notes.)

\bigskip

\begin{exercise}
Let $\left(  a_{0},a_{1},a_{2},\ldots\right)  $ be a sequence of integers
defined recursively by%
\begin{align*}
a_{0}  &  =2,\ \ \ \ \ \ \ \ \ \ a_{1}=3,\\
a_{n}  &  =3a_{n-1}-2a_{n-2}\ \ \ \ \ \ \ \ \ \ \text{for all }n\geq2.
\end{align*}
Prove that $a_{n}=2^{n}+1$ for each $n\in\mathbb{N}$.
\end{exercise}

\bigskip

Now, we recall again the Fibonacci sequence $\left(  f_{0},f_{1},f_{2}%
,\ldots\right)  $ that we got to know in \S 1.5. It is defined recursively by
$f_{0}=0$ and $f_{1}=1$ and $f_{n}=f_{n-1}+f_{n-2}$ for each $n\geq2$.

\begin{exercise}
\textbf{(a)} Let $k\in\mathbb{N}$. Show that
\[
f_{n}^{2}-f_{n+k}f_{n-k}=\left(  -1\right)  ^{n-k}f_{k}^{2}%
\ \ \ \ \ \ \ \ \ \ \text{for every integer }n\geq k.
\]


\textbf{(b)} Which exercise from homework set \#1 does this generalize?
\end{exercise}

\begin{exercise}
Prove that every $n\in\mathbb{N}$ and $m\in\mathbb{N}$ satisfy%
\[
\sum_{k=0}^{n}\dbinom{n}{k}f_{m+k}=f_{m+2n}.
\]

\end{exercise}


\end{document}