I wanted to understand how to solve these types of questions in Combinatorics in which there is a recurrence relation. For example, this question states:  Consider a binary string which consists only of digits 0 and 1, let an be the number of such strings with $n$ digits such that 11 doesn't come anywhere in the string. Find the relation between $a_n, a_{n-1}$ and $a_{n-2}$.  Now I have no idea how to solve this question, any help or solution would be much appreciated.
Find a recurrence relation for the number of $n$-digit binary sequences with no pair of consecutive $1$s.  I know my base case:$$a_1 = 2$$ for either $1$ or $0$. Normally to construct the recurrent relation I would say fill in $n-1$ spaces then add a $1$ or $0$, which is simply:$$a_n = 2\cdot a_{n-1}$$However, I don't believe this is right, because this doesn't take into account that there can't be any consecutive $1$s.  Even when I use the base case of $2$ digits:$$a_2 = 3$$for either $01$, $10$, or $00$, I still get the recurrence relation of:$$a_n = 3\cdot a_{n-2}$$which also does not account for $01$ and $10$ occurring consecutively.   So how would I express this?
Proving the sum of two independent Cauchy Random Variables is Cauchy So I have an old unlocked phone and I'd like to load up my own OS.  I've seen some mods out there.  CyanogenMod seems to be the most popular.  However, I was curious if I could just install a pure untouched version of the OS -- as distributed by Google. I seem to remember being able to download a clean compiled version of the Android OS.    Does Google distribute any binaries like this, and if not, is there someone out there that does?    Additionally, is this even worth trying?  These days is it necessary for Android to be packaged with special drivers and such for all of the different hardware out there? Probability that all three shirts were worn Mr. Jones has three shirts: Red, Green and White. Each day he picks randomly a red shirt with probability of $\frac{1}{2}$, green with probability of $\frac{1}{3}$ and white with probability of $\frac{1}{6}$.     What is the probability that he wears all three shirts after 6 days?   My try:   $$\binom{6}{1}\binom{5}{1}\binom{4}{1}\frac{1}{2}\cdot\frac{1}{3}\cdot\frac{1}{6}\cdot[\binom{3}{2}(\frac{1}{2})^2\cdot[\frac{1}{3}+\frac{1}{6}]+[\binom{3}{2}(\frac{1}{3})^2\cdot[\frac{1}{2}+\frac{1}{6}]+[\binom{3}{2}(\frac{1}{6})^2\cdot[\frac{1}{2}+\frac{1}{3}]+\binom{3}{1}\binom{2}{1}\frac{1}{2}\cdot\frac{1}{3}\cdot\frac{1}{6}]$$  First, we pick possessions for red, green and white. Then, we multiply it by the other combinations possible of the other three possessions left, and summing it up (I didn't count all possibilities). The problem is the answer is bigger than 1, so it must be wrong. Also, I guess there is more elegant way.  Thanks in advance. What does the word treats mean in this context? I don't think it means entertainment, food, drink etc.. Maybe it is an old usage of this word? Does it mean influencing factors or something like that?  This is the first part of the title of the first chapter of the novel "Oliver Twist" by Charles Dickens. The whole title reads:     Treats of the Place Where Oliver Twist was Born and of the Circumstances Attending His Birth No Line Break in Author Command? It goes off the page! I was wondering if there was any was I could right click inside any instance of a "Finder" window such that I have an option that says "Open Terminal Here". It would be really helpful. If a mass moves close to the speed of light, does it turn into a black hole? No "waiting for review" for comment flags Let me explain using screenshot and Free Hand Red Circles and Arrows:        As comment flags are now fully showing in the summary, I would expect the same behavior by showing how many are still active. Let f(t)=2/(t+8)  (a) Simplify the following difference quotient as much as possible: $$[f(t+h)−f(t)]/h$$  MY ANSWER (CORRECT): $-2 / [(t+8)(t+h+8)]$  (b) Use your result from (a) and the limit definition of the derivative to calculate $f′(t)=\lim_{h → 0} [f(t+h)−f(t)]/h=$  MY ANSWER (CORRECT): $-2 / [(t+8)^2]$  (c) Use your answer from part (b) to evaluate $f′(−3)$  MY ANSWER (CORRECT): $-2 / 25$  (d) Find the equation of the tangent line to the curve at the point $(−3,f(−3))$ y=  I have tried the point-slope formula, $y-y_1=m(x-x_1)$, but my answer was incorrect. Finding a 2x2 Matrix raised to the power of 1000 Let $A= \pmatrix{1&amp;4\\ 3&amp;2}$. Find $A^{1000}$.  Does this problem have to do with eigenvalues or is there another formula that is specific to 2x2 matrices? Prove that $\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$ I want to prove the following: $$\sum_{k=1}^{n}k\binom{n}{k}=n\cdot2^{n-1}$$ what I did is(use binominal): $$\binom{n}{k}X^k\cdot 1^{n-k} = (X+1)^n$$ $$k\binom{n}{k}X^k\cdot 1^{n-k} = k(X+1)^k-1$$ now I replace $k$ by $n$ and insert $X=1$ , I will get now $n\cdot2^{n-1}$ what you are suggesting? there is another way to do that? thanks! How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$? I have 28,000 files of size 6.7 GB in the directory aaa/input. I need create TAR files in the directory aaa/output, such that each TAR file cannot exceed 1.9 GB, so I have to tar the 28,000 files into 4 TAR files (6.7/1.9) as Log_(current date)_01.tar Log_(current date)_02.tar ... etc.  such that each tarball is less than 1.9 GB. Is there a way to do this using a Unix script? Problems with FOUC Stop jQuery .load response from being cached I have the following code making a GET request on a URL:  $('#searchButton').click(function() {     $('#inquiry').load('/portal/?f=searchBilling&amp;pid=' + $('#query').val());             });   But the returned result is not always reflected. For example, I made a change in the response that spit out a stack trace but the stack trace did not appear when I clicked on the search button. I looked at the underlying PHP code that controls the ajax response and it had the correct code and visiting the page directly showed the correct result but the output returned by .load was old.  If I close the browser and reopen it it works once and then starts to return the stale information. Can I control this by jQuery or do I need to have my PHP script output headers to control caching? What do I do if I don't have a META-INF folder or minecraft.jar? I have looked all over the Internet for solutions but I have got nothing.  I want to install the minepainter mod but I need a META-INF folder and a minecraft.jar. Compute $E(X\mid X+Y)$ for independent random variables $X$ and $Y$ with standard exponential distributions Being $X$ and $Y$ independent random variables distributed as $\mathrm{Exponential}(1)$ and let $T=X+Y$ Calculate $E(X|T)$.  This is my attempt at solving this:  Being $Z=\sum_{i=1}^{n}P_{i}$ where $P_{i}$~$\mathrm{Exp}(\lambda)$ then using it's MFG you have that $$M_{P_i}(t)=\frac{\lambda}{\lambda-t}$$ then $$M_{Z}(t)=M_{\sum{P_{i}}}(t)=(M_{P_{1}})...(M_{P_{n}})=(\frac{\lambda}{\lambda-t})^n$$ with this being the MFG of a $\mathrm{Gamma}$ distribution with parameters $(n,\lambda)$  With this result is easy to see that $T$~$\mathrm{Gamma}(2,1)$  Now $E[X|T]=\int xf_{X|T}(x|t)dx$  Then using Bayes' theorem we can make $$f_{X|T}(x|t)=\frac{f_{T|X}(t|x)f_{X}(x)}{f_{T}(t)}$$ then $$f_{T|X}(t|x)=Pr(T=t|X=x)$$ $$=Pr(X+Y=t|X=x)$$ $$=Pr(x+Y=t|X=x)$$ $$=Pr(Y=t-x)=f_{Y}(t-x)$$  Using this result we can now rearrenge $$f_{X|T}(x|t)=\frac{f_{Y}(t-x)f_{X}(x)}{f_{T}(t)}$$ $$=\frac{e^{-(t-x)}.e^{-x}}{\frac{te^{-t}}{\Gamma(2)}}$$ $$=\frac{1}{t}$$ then $$E[X|T]=\int (x)(\frac{1}{t})dx$$  And this where I got stuck because $1_{x\epsilon(0,\infty})$, I made this explanation step-by-step hoping that someone can se easily check where I went wrong. I would aprecciate a step-by-step answer, thanks. Stack Overflow is a site of passers-by They says that Stack Overflow is a community-driven web-site. But, as a matter of fact, it is not.  There is no community at all. It's more like an imageboard, as a number of participants makes them pretty much like anonymous to each other despite of registration.   Stack Overflow is a site of passers-by.  Passers-by who want to solve their silly typo problem and passers-by who eager to share their vast knowledge in spotting silly typos. And - oh - sometimes to perform a civic duty and vote to close a question they have no idea what to answer to and thus calling it "not a real question".  So, as a funny consequence, whatever question can get   answered upvoted downvoted closed deleted   depends not on its quality but only on the fluctuation happened to "community" at the given moment, at a pretty random rate.  It is not only quite discouraging. It makes the very idea of collecting and improving knowledge quite impossible. How to prove that for all natural numbers, $4^n > n^3$? This is a problem set I have, it's not a homework but it's very important practice...  Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably...   The problem is:     Prove that for all natural numbers $n$,  $4^n &gt; n^3$ My Folders Become Hidden System Files And Access Denied? I have an external hard disk , just connected to another system and when i connected to my own computer one of my folders which contain photos become hidden system files AND all folders in it rename to same name !?!? and my access is denied ( Screenshot )    I tried to change ownership but i can't  Any solution !?