Fibonacci numbers $f(n)$ are defined recursively:  $f(n) = f(n-1) +f(n-2)$ for $n &gt; 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt 5}{2}\right)^n- \left(\dfrac{1 - \sqrt 5}{2}\right)^n \tag1$$  How to prove (1) using induction?  Remarks One could get (1) by the general method of solving recurrences: look for solutions of the form $f(n)=r^n$, then fit them to the initial values. But there should be a more concrete proof for this specific sequence, using the principle of mathematical induction.
This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n &gt; 1$,  where $u_0 = 0$ and $u_1 = 1$.   Show that     $$u_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}.$$    Please help me with its proof. Thank you.
Power series is locally normally convergent in its convergence radius $B(0,R)$ Tikz: Arrowheads in the center When can we find holomorphic bijections between annuli? I am using ArcMap 10.1. The background: I have an .mxd that I intend to roll out across the company as our default template for all future maps/figure production. This .mxd contains two Data Frames; the first (&quot;Main&quot;) is larger (occupying almost the entire page) and is designed to be used to hold the majority of mapped data whilst the second (&quot;Inset&quot;) is much smaller and is to be used as an overview map. The problem: When choosing to import this new .mxd to an existing .mxd (using the Change Layout button on the Layout toolbar) the wizard automatically selects the smaller (&quot;Inset&quot;) Data Frame to occupy the larger part of the page. The screenshot below shows this interface with a red box indicating the two possible locations of each Data Frame:  Whilst it is possible to manually alter the order of the Data Frames using the Move Up or Move Down buttons, how do I change the default option for this setting (i.e. so that the &quot;Main&quot; Data Frame always appears in the larger space - &quot;2&quot;)? The reasoning is that I would like the process of using this new .mxd template to be as simple as possible, this is especially important as we have a wide range of users with varying abilities. Does anyone know how to choose/change this default setting? It does not appear to be in the Data Frame Properties &gt; Data Frame menu where I would expect it to be. Is there a convenient way to import a schema into Oracle 11gR2 using a single new or different tablespace than where the data originated?  As an example, I have exported BLOG_DATA from OLDDB, where all user data is stored in the USERS tablespace.   On NEWDB, I would like to import the BLOG_DATA schema, but store the user objects in the BLOG_DATA tablespace, created specifically for this user.   I have created the BLOG_DATA user, created the BLOG_DATA tablespace and set it as the default tablespace for that user and added an appropriate unlimited quota.  CREATE TABLESPACE blog_data DATAFILE SIZE 1G;  CREATE USER blog_data IDENTIFIED BY secretpassword DEFAULT TABLESPACE blog_data QUOTA UNLIMITED ON blog_data;  GRANT connect,resource TO blog_data   The schema was exported from OLDDB with something like  exp blog_data/secretpassword@OLDDB file=blog_data.dmp    After reading Phil's excellent answer below, I found myself wondering:   Since data has no other place to go than the default tablespace - the only tablespace the user has a quota on - will this effectively force imp to put all user objects in that default tablespace?  imp blog_data/secretpassword@NEWDB file=blog_data.dmp   Would this then place the entire blog_data schema in the blog_data tablespace on NEWDB? Is there any reason why this wouldn't work or that I would run into problems with certain objects etc?  update:  I did a quick test and found this to be the case. Imp places objects in the default tablespace for that user, provided it can't place it in the original tablespace (eg. the tablespace does not exist). Full explanation:   Still, I suppose that using Data Pump like Phil suggests might be the preferred option. In what cases does the gerund stemming from a transitive verb take the direct object of that verb and when is a prepositional complement used?  For example:    Brown's deft painting of his daughter is a delight to watch (it is a delight to watch while Brown deftly paints his daughter). Painting his daughter is an outlet for his energy. Securing safety for prisoners is the main problem for the penitentiary system. The widely discussed problem of the securing of prisoners' safety was given priority. Inconsistent behavior when typing a number into the search field Adding only specific text from a file to an array list, part 2 - Java Is the external micro SD card path the same for all devices? I am using a Samsung Tab 2.  There is a path: /mnt/external_sdcard/myfile. Here, I can save, read, or write my file to the external micro SD card.  I'm just wondering if this path is the same for all devices, or if it may be different. Plotting complex numbers as an Argand Diagram I think the title is pretty self explanatory.   The question is how can I create an alias in a Windows cmd that will also work after I close and then reopen it. For example I would like an alias called ip to call the command ipconfig or an alias ls -l to call Dir.  My operating system is Windows 10, in case that makes a difference. Is it ethical to withdraw a paper after acceptance in order to resubmit to a better journal? A ring element with a left inverse but no right inverse? Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of the determinants, and that implies that no faithful finite-dimensional representation must exist, and my imagination seems to have given up on me :) What would be the nicest proof of the following theorem:  If $\lim\limits_{n \rightarrow \infty} a_{n} = \infty$, then $\lim\limits_{n \rightarrow \infty} \left(1 + \frac{1}{a_{n}} \right) ^ {a_{n} } = e$. If $\lim\limits_{n \rightarrow \infty}b_{n} = 0$, then $\lim\limits_{n \rightarrow \infty} \left(1 + b_{n} \right) ^ {\frac {1} {b_{n}} } = e$.  I somehow failed to find a proof here on the website and in the literature. Is the Nintendo 3DS region-locked? I am thinking of buying a 3DS when the price drop comes into place in America, however I live in Australia. Will Australian 3DS games work on a North American 3DS? What are the disadvantages of using a Zener diode over a linear voltage regulator? I wonder why a Zener diode is worse than linear voltage regulator.  Some of the posts online mentioned that because of the power dissipating problem. Zener diodes keep consuming a large amount of current while the load may just need a very little amount of current.   If I understand correctly, this phenomenon will happen only when the Zener diode is connected with the load in parallel. What if I connected my Zener diode with my load in series? Also, someone said that linear voltage regulators (e.g. LM7805) are not good voltage regulators as the power dissipated in them is much higher than that in switching voltage regulators. In my understanding, the power dissipated in a linear voltage regulator can be calculated by dropout voltage x current = power converted to heat.   Regarding these 2 points, other than a linear voltage regulator can handler a wider current range, what is the benefit of choosing a linear voltage regulator over a Zener diode? How do I get AMD/Intel Hybrid Graphics drivers to work? On Ubuntu 12.04 my Sony Vaio VPCSE with Intel® HD Graphics 3000 and AMD Radeon™ HD 6630M worked fine with the Catalyst Control Center version 12.6. Also the switching between integrated and discrete graphic card was working.  In both cases, I followed this .  But it is not working on Ubuntu 12.10.   I tested the tutorial with the Catalyst Control Center from the Ubuntu Software Center, the version 12.8 and version 12.9. Always the same problem: After installation process I am able to boot in the login-screen. But after entering the password for my username, only the background-image appears. Unity seems not to be starting. I am only able to reach the context-menu by right clicking the mouse button.  I also tried , but is also does not work for me.  Any ideas what to do to fix this problem?   Update: Same problem on Ubuntu 13.04!  Update: Solution for 13.04 ist not working on Ubuntu 13.10! In-Page Editor Line Distortion The SFDC In-page editor has always had distortion for me, and I'm just wondering if it's just a lousy browser plugin they're using or if someone has found a good browser / operating system combination that stops the obnoxious distortion.   Obviously Salesforce doesn't have much interest in fixing it: as it's been this way on multiple operating systems and browsers I've used since I've used Salesforce.  However, my frustration with their other developer tools has lead me to use it occasionally and it's always the same: failed to fetch security undermined error [IP: 91.189.91.38 80] in ubuntu 20.04 Persistent Dummy Output The soundcard I want to use is detected and works (it only recently switched to dummy output and didn't allow me to go back). HDMI is no longer listed under Audio and Video - KDE Control Module; everything under audio module is set to dummy output and its the only 'soundcard' on the list now. I can view my previous sound cards in alsamixer, but changing the settings from there does nothing. How can I get my real soundcards to work again? I have already tried the following answers without success:  EDIT: name@name-desktop:~$ lspci -vvv | grep -A8 Audio 00:01.1 Audio device: Advanced Micro Devices, Inc. [AMD/ATI] Kaveri HDMI/DP Audio Controller         Subsystem: ASRock Incorporation Kaveri HDMI/DP Audio Controller         Control: I/O+ Mem+ BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR- FastB2B- DisINTx+         Status: Cap+ 66MHz- UDF- FastB2B- ParErr- DEVSEL=fast &gt;TAbort- &lt;TAbort- &lt;MAbort- &gt;SERR- &lt;PERR- INTx-         Latency: 0, Cache Line Size: 64 bytes         Interrupt: pin B routed to IRQ 43         Region 0: Memory at ff764000 (64-bit, non-prefetchable) [size=16K]         Capabilities: &lt;access denied&gt;         Kernel driver in use: snd_hda_intel         Kernel modules: snd_hda_intel -- 00:14.2 Audio device: Advanced Micro Devices, Inc. [AMD] FCH Azalia Controller (rev 01)         Subsystem: ASRock Incorporation FCH Azalia Controller         Control: I/O- Mem+ BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR- FastB2B- DisINTx-         Status: Cap+ 66MHz- UDF- FastB2B- ParErr- DEVSEL=slow &gt;TAbort- &lt;TAbort- &lt;MAbort- &gt;SERR- &lt;PERR- INTx-         Latency: 32, Cache Line Size: 64 bytes         Interrupt: pin A routed to IRQ 16         Region 0: Memory at ff760000 (64-bit, non-prefetchable) [size=16K]         Capabilities: &lt;access denied&gt;         Kernel driver in use: snd_hda_intel  Could this image possibly help?:  There is another soundcard but I'm unsure of what its for because it never outputs sound, but it does have a lot of settings: that sound card is labelled as &quot;HD-Audio Generic&quot;.   So I went into Software &amp; Updates under the Additional drivers tab and I see this:  It was on Do not use the device so I changed it and applied the settings but unfortunately the soundcard isn't back even after reboot.