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\begin{document}

\begin{center}
{\huge  \sc   Homework 6} \\
\smallskip
{ Due Date: 11/02/2021}\\
\smallskip
{Name:  }\\
{\large  }
\end{center}
\vspace{1cm}

\begin{enumerate}

\item[8.3] 
\begin{enumerate}
\item Work out 1, 1 + 8, 1 + 8 + 27 and 1 + 8 + 27 + 64. Guess a formula

$\sum_{r=1}^nr^3 $
and prove it.

\newpage
\item Check that 1 = 0+1, 2+3+4 = 1+8 and 5+6+$\ldots$+9 = 8+27.
Find a general formula for which these are the first three cases. Prove
your formula is correct.

\end{enumerate}
\newpage
\item[8.6] The Lucas sequence is a sequence of integers $l_1,l_2,...,l_n,...,$ such that
$l_0=2, l_1 = 1,l_2 = 3$ and
\[
l_{n+1} = l_n +l_{n-1}
\]
for all $n \geq 1$. So the sequence starts 1,3,4,7,11,18,...
Find the pattern for the remainders when $l_n$ is divided by 3. (Hint: Consider the first 8 remainders, then the next 8, and so on; formulate a conjecture for the pattern and prove it by induction.)
Is $l_{2000}$ divisible by 3?
\newpage

\item[8.9] 

Prove that if $0 < q <\frac{1}{2}$ then for all $n \geq 1$,
\[
(1+q)^n \leq 1+2^nq
\]

\newpage

\item[8.11] Just for this question, count 1 as a prime number. A well-known result in
number theory says that for every integer $x \geq 3$, there is a prime number
$p$ such that $\frac{x}{2} < p < x$. Using this result and strong induction, prove
that every positive integer is equal to a sum of primes, all of which are
different.

\newpage

\item[8.13]

\begin{enumerate}

\item  Suppose we have $n$ straight lines in a plane, and all the lines pass
through a single point. Into how many regions do the lines divide the
plane? Prove your answer.
\newpage 
\item  We know from Example 8.8 that n straight lines in general position
in a plane divide the plane into $\frac{1}{2}(n^2+n+2)$ regions. How many of
these regions are infinite and how many are finite?
(In case of any confusion, a finite region is one that has finite area; an
infinite region is one that does not.)

\end{enumerate}
\end{enumerate}

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