Problem   Find generalitazion for matrix A exponents, when $n\in\{1,2,3,\dots\},n \in \mathbb{N}$    $$A^n=\begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{bmatrix}^n , \text{ when } n \in \mathbb{N}$$  Proof generalization by induction.   Attempt to solve   By computing a set of $A$ exponent's $n\in \{\ 1,2,3,4 \}$. It is possible  to form generalization that is applicable for set $n\in \{1,2,3,4 \}$  $$ A^1=\begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{bmatrix},A^2=\begin{bmatrix} 1 &amp; 2 \\ 0 &amp; 1 \end{bmatrix},A^3=\begin{bmatrix} 1 &amp; 3 \\ 0 &amp; 1 \end{bmatrix},A^4=\begin{bmatrix} 1 &amp; 4 \\ 0 &amp; 1 \end{bmatrix} \dots A^n =\begin{bmatrix} 1 &amp; n \\ 0 &amp; 1 \end{bmatrix} $$   Induction proof   Induction hypothesis  Assume expression is valid when $n=k$  $$ A^k = \begin{bmatrix} 1 &amp; k \\ 0 &amp; 1\end{bmatrix} $$  Base case  When $n=1$ $$ A^1=\begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{bmatrix} $$ which is valid by definition.   Induction step  When $n=k+1$  $$ A^{k+1} = \begin{bmatrix} 1 &amp; k+1 \\ 0 &amp; 1\end{bmatrix}$$  $$ A^{k+1}=A^kA^1 $$  By utilizing induction hypothesis we have  $$ \implies A^{k+1}=\begin{bmatrix} 1 &amp; k \\ 0 &amp; 1\end{bmatrix}\begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{bmatrix} $$ By utilizing matrix multiplication we have $$ \implies A^{k+1} \begin{bmatrix} 1\cdot 1 + k\cdot 0 &amp; 1 \cdot 1 + 1 \cdot k \\ 0 \cdot 1 + 1 \cdot 0 &amp; 0 \cdot 1 + 1 \cdot 1 \end{bmatrix} $$ $$ \implies A^{k+1}=\begin{bmatrix} 1 &amp; k+1 \\ 0 &amp; 1 \end{bmatrix} $$  $$ \tag*{$\square$} $$     EDIT   The point of posting this was to have comment on if my solution seems correct or not. If you can notice something that doesn't look right, let me know !
Well, the original task was to figure out what the following expression evaluates to for any $n$.  $$\begin{pmatrix} 1 &amp; 1 \\ 0 &amp; 1\end{pmatrix}^{\large n}$$  By trying out different values of $n$, I found the pattern: $$\begin{pmatrix} 1 &amp; 1 \\ 0 &amp; 1\end{pmatrix}^{\large n} = \begin{pmatrix} 1 &amp; n \\ 0 &amp; 1\end{pmatrix}$$  But I have yet to figure out how to prove it algebraically.   Suggestions?
How to install Internet Explorer (multiple versions)? I am developing one site which has chat feature in Rails. I am using Ubuntu 12.04 LTS. I need to check that the site is IE9 because the chat feature using socket io of HTML5. I want to know that how to install IE9 in Ubuntu. I have looked at Chrome and Firefox extensions but if any standalone is available then please give me suggestion. Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.   For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.  Is there a general method which allows us to evaluate the indefinite integral     $$ \int P(\sin x, \cos x) \text{d} x$$   What about the case when $\displaystyle P(x,y)$ is a rational function (i.e. a ratio of two polynomials)?   Example of a rational function: $\displaystyle \frac{x^2y + y^3}{x+y}$.    This is being asked in an effort to cut down on duplicates, see here:   and here: Non-normality in residuals Getting "'file' is different from the same file on the system" when I attempt to update through synaptic Encountered an error while updating through Synaptic Manager. Subsequent updates gave the error at the bottom of the output. I'm a noob so any help is appreciated.  al@al:~$ sudo apt-get -f upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following NEW packages will be installed:   libjack-jackd2-0:i386 The following packages have been kept back:   linux-headers-generic skype 0 upgraded, 1 newly installed, 0 to remove and 2 not upgraded. Need to get 0 B/202 kB of archives. After this operation, 493 kB of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 310056 files and directories currently installed.) 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