Question

If the product of $n$ matrices $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ 0& 1\end{array}\right]\dots$ $\left[\begin{array}{cc}1& n\\ 0& 1\end{array}\right]$ is equal to the matrix $\left[\begin{array}{cc}1& 378\\ 0& 1\end{array}\right]$ then the value of $n$ is equal to

Answer 1

Answer: 27

[Product $=\left[\begin{array}{cc}1& \frac{n(n+1)}{2}\\ 0& 1\end{array}\right]$
$p=\left[\begin{array}{cc}1& 1+2+3+\dots .+n\\ 0& 1\end{array}\right]$
$\therefore \frac{n(n+1)}{2}=378$
$\Rightarrow n=27$