I want to calculate the determinant of the following matrix $$\begin{pmatrix} \alpha +\beta&amp;\alpha\beta&amp;0&amp;\cdots&amp;0&amp;0\\ 1&amp;\alpha + \beta &amp;\alpha \beta&amp;\cdots&amp;0&amp; 0\\ 0&amp;1&amp;\alpha + \beta &amp; \cdots &amp;0&amp;0\\ .&amp;.&amp;.&amp;.&amp;.&amp;.\\ .&amp;.&amp;.&amp;.&amp;.&amp;.\\ .&amp;.&amp;.&amp;.&amp;.&amp;.\\ .&amp;.&amp;.&amp;.&amp;.&amp;.\\ 0&amp;0&amp;0&amp;0&amp;1&amp;\alpha + \beta\end{pmatrix}$$  Here is my attempt    Is this correct?  I used Laplace
Given this (very) tricky determinant, how can we calculate it easily?  $$\begin{pmatrix} \alpha + \beta &amp; \alpha \beta &amp; 0 &amp; ... &amp; ... &amp; 0 \\ 1 &amp; \alpha + \beta &amp;  \alpha \beta &amp; 0 &amp; ... &amp; 0 \\ 0 &amp; 1 &amp; \alpha + \beta &amp; \alpha \beta &amp; ... &amp; ... \\ ... &amp; ... &amp; ... &amp; ... &amp; ... &amp; 0 \\ ... &amp; ... &amp; .... &amp; ... &amp; ... &amp; \alpha \beta \\ 0 &amp; 0 &amp; 0 &amp; ... &amp; 1 &amp; \alpha + \beta   \\ \end{pmatrix} \in M_{n\times n}$$  EDIT:     I have to prove it is equal to $\frac{{\alpha}^{n+1} - {\beta}^{n+1}}{\alpha - \beta}$   Any help is appreciated, I just could not find a trick to ease it up!
Continuity $\iff$ Preimage is closed whenever set is closed I'm trying to do the following:     Prove that a function $f : S\rightarrow T$ between two topological spaces is continuous iff $f^{-1}(C)$ is closed whenever $C\subset T$ is closed.   To show that the closed-set relationship implies continuity, notice that $T-C$ is open. The preimage of this open set is $f^{-1}\left(T-C\right)=S-f^{-1}(C)$, which is open by assumption ($f^{-1}(C)$ is closed). Of course, any open set $G\subset T$ can be written in the form $T-C$ by letting $C=T-G$. Thus, for every open set $G\subset T$, the preimage is open.  To show the implication in the other way, consider a closed set $C\subset T$. $T-C$ is open, so $f^{-1}(T-C)=S-f^{-1}(C)$ is open as well ($f$ is continuous), which means that $f^{-1}(C)$ is closed. Integrate $\int_{0}^{\pi}{\frac{x\cos{x}}{1+\sin^{2}{x}}dx}$ Why AccessViolationException cannot be caught by .NET4.0 Unbounded continuous function on non-compact metric space Suppose $(S,d)$ is a non-compact metric space. Is it possible to construct an unbounded continuous function from $S$ to $\mathbb{R}$? If it is possible, please show the construction method :)  Here's my attempt to solve this problem: Because $S$ is not compact, we can find a neighborhood assignment function $N(x)$ so that a finite cover is not possible. Then each time we can find an uncovered point from the union of neighborhoods before, and change the function by $f(x):=max\{f(x),\text{a function which equals to the indice of this point at that point and gradually vanishes at further points from the selected point}\}$. However I'm concerned that while this function is unbounded, there may be some invalid points for this function (goes to infinity)...  EDIT: As I found out there is a duplicate question, however the elementary solution is not presented in that question, so I still wish to know about this problem. :) How do I end a Tkinter program? Let's say I have this code:  from Tkinter import *  def quit():     # code to exit  root = Tk() Button(root, text="Quit", command=quit).pack() root.mainloop()   How should I define the quit function to exit my application? Reference undefined in \section with amsbook I have a section that is devoted to the proof of a theorem, and hence I used  \section{Proof of Theorem~\ref{thm}}   Of course, because of the headers that are capitalized, it asks for \ref{THM} when producing the header, and I get an obvious error. What is the proper way to handle such a case? 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For step $1$: if $a$ is a member of $A$ then $a = (n-1)/(2n+3)$ for some natural number $n$ and $\frac{n-1}{2n+3} \leq \frac{n}{2n} = \frac12$ so $\frac12$ is an upper bound for the set  Step 2 is where I am uncertain. I think I need to show that for any $y&lt;\frac12$ there exists a member of the set between $y$ and $\frac12$ but not sure how to do this. Thanks for all the answers so far! Spurious @ symbol at start of bash prompt in Debian Jessie Unable to Change Apple ID and password I have tried repeatedly to change my Apple ID and password without success. I have followed all of the directions trying both my emails and asked for new passwords----no luck whatsoever. How to properly hook up various maps types in Cycles? I created some texture maps in MindTex to use in Blender, but I can't figure out how to properly hook them up. 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Can I install any iOS version I wish by picking and choosing which firmware to restore? I'm trying to downgrade my iPhone from the current iOS to an older one, as per .  But every time I try, I always receive this error:     The iPhone could not be restored. This device isn't eligible for the requested build.     I have tried many times with and without DFU mode. Any ideas regarding this? Is there a fast way to get to the bottom of the Death Star? In Tiny Death Star, you can get to the top of your tower by tapping on the top of the screen while avoiding the "Add Level" button. However, the only way I can find to go to the bottom of the tower is to wait for someone to want an elevator ride, and tap on the button to give them that ride. Is there any other hot-spot for getting to the bottom of your Death Star? Why does no physical energy-momentum tensor exist for the gravitational field? 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