If $\{ a_n\}$ be a sequence of positive real numbers such that $$a_n=\sqrt{a_{n-1}a_{n-2}},n&gt;2$$ then show that the sequence is convergent to $(a_1a_2^2)^{1/3}$.    How I proceed: Take square of $a_n$ so I get $a_n^2=a_{n-1}a_{n-2} \implies a_n^3=a_na_{n-1}a_{n-2}$. Then stuck. Please help.
We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit.
How to register domain for 15 years? I have heard that Google trust that website which has long expiry date. that want to know how can I register my domain for a long period. How to keep font preferences in TexWorks 4 The default font is too small on my screen at the resolution I have it, so I like to reset the font to size 12, but it appears I have to do this everytime I launch the program.  How do I make it remember my preferences? May I summarize all answers on StackOverflow, I asked a . There are many good answers but I can not accept just one of it. I would prefer to create my own answer where I summarize all answers. Is that allowed? My company has purchased a product that renders an ASP.NET control on the page. This control uses jQuery 1.2.3 and adds a script tag to the page to reference it. The developers of the control will not support use of the control if it modified in any way (including modification to reference a different version of jQuery).  I'm about to start development of my own control and would like to use the features and speed improvements of jQuery 1.3. Both of these controls will need to exist on the same page.  How can I allow the purchased control to use jQuery 1.2.3 and new custom development to use jQuery 1.3? Also out of curiosity, what if we were to use an additional control that needed to reference yet another version of jQuery? Make Blender usable on high resolution screen Is it safe to use the hash of the data as the key to encrypt them? Is it safe to use the hash of the data as the key to encrypt them?  That is, if I give one some encrypted data and tell them that the key that was used to encrypt them was the hash digest of the data before encrypting, will it be easier for them to decrypt them?  Edit as I can not reply to comments: The recipient will have the keys, that is no problem. Ubuntu Server 20.04 Nvidia Driver Dependencies I had a hard time installing Nvidia drivers in a fresh install of Ubuntu 20.04.  Installing via sudo apt install nvidia-driver-430 resulted in a list of 542 packages to install.  I ended up spending a bit of manual effort to get the Nvidia-provided driver installed without nearly so many dependencies (only had to install gcc and make).  Is this massive number of dependencies normal for Ubuntu Server?  I did a very vanilla install without the GUI.  Reddit suggested I ask here, as their opinion is that I should just install from the default repos, but I felt that so many packages was absurd just to install a driver.  Edit for clarity: installing the nvidia-driver-430 package appears to install the entire Gnome shell, which I feel is unnecessary on a server. Will adding more RAM speed up my computer? Apparently, you can speed up your computer by adding more RAM. Is the amount of ram to be added is directly or indirectly related to the amount of hard-disk  space, processor speed, Motherboard etc... Can the memory of a RAM be compared with Virtual Memory and Cache Memory?  Note: About my PC Configuration: Intel core i5 processor, Motherboard DP55WB 4GB RAM, 700 GB Hard-disk space. A worker carries two bags. Each of the bags initially contains N nails. Whenever the worker needs a nail, he takes it from a bag picked at random. At some point the worker picks an empty bag. Find the probability that the other bag contains exactly m nails.  My reasoning:  The desired probability: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$, where A, B - sets of nails in each bag. Then we can "divide" this probability (I suppose that I deal with independent events):  $P(|A| = 0\ and\ |B| = m) = P(|A| = 0) \cdot P(|B| = m);$  $P(|A| = 0) = (\frac{1}{2})^N$, where $|A| = |B| = N$ (initial condition).  $P(|B| = m) = (\frac{1}{2})^{N - m}$  Result: $C_2^1 \cdot P(|A| = 0\ and\ |B| = m)$ = $C_2^1 \cdot (\frac{1}{2})^{2N - m}$  Am I wrong or it is correct? So I am asked to find all the isomorphisms between $G = (\mathbb Z_4,+)$ and $H = (U_5,*)$.  I solved it as follows: we will have two isomorphism corresponding to the two generators of $U_5$.  The first isomorphism : $\phi([x]_4)$ = $[2]_5^x$   The second isomorphism is: $\phi([x]_4)$ = $[3]_5^x$   Notice that [2] and [3] are the only generators of $U_5$ one can check it systematically or use the fact that $a^m$ is a generator iff gcd(m,4) = 1.  So I just wanted a verification of my work I showed that they are bijective, well defined, and is a homomorphism. When a ball moves to the right, friction acts to oppose the motion, in other words, to the left. However, when a car travels around a bend, the friction acts in the perpendicular direction to the car's velocity and provides the centripetal force. I just cannot understand why friction would act in that direction. Turning off swapping for only one process with cgroups? I would like to turn off swapping for only one process. Swap should work as usual for the other processes.  How I can implement this with cgroups? How to deal with high correlation among predictors in multiple regression? Help with footer always to bottom Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$ Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$.   My try:  Let $\mathbb Z^n\cong \mathbb Z^m $. To show that $m=n$.  Case 1: Let $m&gt;n$. Now that $\mathbb Z^m$ has $m$ generators whereas $\mathbb Z^n$ has $n$  generators and an isomorphism takes a generator to generator ; that is the contradiction.  Please correct me if I am wrong.  The case  $m&lt;n$ also follows similarly. How to boot from non-Ubuntu live ISO images like Fedora or CentOS? Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$ Find the order of $8/9, 14/5,48/28$ in the additive group of $\mathbb{Q}/\mathbb{Z}$    JUNK  Guessing asking for $8/9 * k = 0 $ (being a whole integer) for smallest integer so for this one is 9 . another guess would be the smallest element of $Q/Z$ being something like $9/8$  $14/5$ would be 5  $48/28=12/7$ would be 7? Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$ This an interesting problem my friend has been working on for a while now (I just saw it an hour ago, but could not come up with anything substantial besides some PMI attempts).  Here's the full problem:  Let $x_{1}, x_{2}, x_{3}, \cdots x_{y}$ be all the primes that are less than a given $n&gt;1,n \in \mathbb{N}$.   Prove that $$x_{1}x_{2}x_{3}\cdots x_{y} &lt; 4^n$$  Any ideas very much appreciated! Electric Potential produced by Hollow Sphere Let there be a hollow sphere (Throughout the question we will ignore the thickness of the sphere) which is positively uniformly charged,q of radius,  R. Suppose there to be a test positive charge, q' at point P which is r distance away from the hollow sphere(r>R, P is outside of hollow sphere).  Now I have an doubt on evaluating Electric Potential,V at point P.  Both source charge,q and test charge,q' are +ve, so source charge will repel the test charge which will make the charge, q' to go to infinity.I have to apply external force, F' to displace the charge, q' from infinity to point P against the equal but repulsive force, F.In short, F' is  displacing the q' from infinity to P.  So External Force, F' will do +ve work and Repulsive Force, F will do -ve work. This way the Electric Field,E produced by hollow sphere is Opposite to Displacement of test charge(i.e.,θ=180).  Now we know that E=k(q/(r^2))  [Please Notice that infinity is the lower limit of integration]  Also V=-∫r∞(E dr cosθ)=-∫r∞(E dr cos 180)=∫r∞(E dr)  At last I ended up to V=- k(q/r). I know that V=k(q/r).  I just want to know why I am having this shitty negative sign in my equation. If particles are points, then aren't atoms empty space? Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing?