Question

If the product of $n$ matrices $\begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & 2 \\ 0 & 1\end{bmatrix}\begin{bmatrix}1 & 3 \\ 0 & 1\end{bmatrix} \ldots$ $\begin{bmatrix}1 & n \\ 0 & 1\end{bmatrix}$ is equal to the matrix $\begin{bmatrix}1 & 378 \\ 0 & 1\end{bmatrix}$ then the value of $n$ is equal to

Answer 1

[Product $=\begin{bmatrix}1 & \frac{n(n+1)}{2} \\ 0 & 1\end{bmatrix}$
$p=\begin{bmatrix}1 & 1+2+3+\ldots .+n \\ 0 & 1\end{bmatrix}$
$\therefore \frac{n(n+1)}{2}=378$
$\Rightarrow n=27$