Solve the system of $n$ equations and $n$ unknowns which the equations are $\left (L_{i} \right ): x_{i}-x_{i+1}=1$ for $i=1,2,....,n-1$ and $\left (L_{n} \right ):x_{n}=1$
$\left\{\begin{matrix} x_{i} - x_{i+1}=1\\ x_{n}=1 \end{matrix}\right.$
$\left\{\begin{matrix} x_{1} - x_{2}=1\\ x_{n}=1 \end{matrix}\right.$
$\left\{\begin{matrix} x_{2} - x_{3}=1\\ x_{n}=1 \end{matrix}\right.$
$\vdots $
$\left\{\begin{matrix} x_{n-1} - x_{n}=1\\ x_{n}=1 \end{matrix}\right.$
$\left\{\begin{matrix} x_{n-1} =2\\ x_{n}=1 \end{matrix}\right.$
solution is $\left ( 2;1 \right )$
is it correct guys ? Thanks for looking
Then, we will also have $$x_{n-2}-x_{n-1} = 1\Rightarrow x_{n-2} = 3$$Similarly $$x_{n-3}-x_{n-2} =1 \Rightarrow x_{n-3}=4$$ $$\vdots$$ $$ x_1-x_2 =1\Rightarrow x_1 = n $$ as we are supposed to find a solution for all $i $.
Hope it helps.