final_quiz_2017.tex

\yourname

\activitytitle{Final quiz}{100 points, 20 for each problem}

\noindent
Write your name only on this page.
Write your solutions on the blank sheets of paper, but do not write your name on them.
When you are done, I will staple this cover sheet to your solutions.

\noindent
Use good form in every problem.
That is, be very clear about every step of the proof, especially things like ``Let $n \geq 1$.'' and generalizing.
This is an extremely important part of this quiz.

\noindent
If you are totally stuck, you can ask for a hint, but it may cost you some points.
Good luck!

\vspace{0.2in}

\blist{0.9in}
\item a) Use mathematical induction to show that $9^n - 8n -1$ is a multiple of 64 for each integer $n \geq 1$.

b) Explain how you have shown that $P(n)$ being true implies that $P(n+1)$ is true {\bf for all $n \geq 1$}.

\item Suppose that $x \leq 7 + \frac{1}{\sqrt{n}}$ for all integers $n \geq 1$.
Show that $x \leq 7$.

\item a) Suppose that $A_n \subseteq B$ for all $n \geq 1$.  Show that $\displaystyle \bigcup_{n=1}^{\infty} A_n \subseteq B$.

b) Suppose that $\displaystyle \bigcup_{n=1}^{\infty} A_n \subseteq B$.
Show that for all $n \geq 1$, we have $A_n \subseteq B$.

\item Show that $\displaystyle \bigcap_{n=1}^{\infty} (3,7+\frac{1}{\sqrt{n}}) = (3,7]$.

\item Show that for all odd integers $n \geq 1$, the expression $n^3 - 25n$ is a multiple of 24.
Actually, we already know this is true, because we showed that $n^3-n$ is a multiple of 24, and we can write $n^3-25n = n^3-n-24n$.  But here I want you to write a new proof that does not use the result that $n^3-n$ is a multiple of 24, and does not simply repeat that proof and then subtract $24n$.

I recommend that you either try factoring $n^3-25n$ and use an approach like we did in class, or else try induction.
Be patient, you can do this, but it will take some time either way.

\elist