We are given a system with n linear equation:
$$\forall i\in \{1,...,n\}: i \cdot x_i + \sum_{j=i+1}^{n}x_j= \frac{i}{n}$$
Prove that the solution for this system of equation is $$\forall i\in \{1,...,n\}: x_i = \frac{1}{n}\left(1-\sum_{j=i}^{n-1}\frac{1}{j}\right)$$
My idea so far:
I wrote the system of linear equations as a matrix first, resulting in
$$Ax = \begin{pmatrix} 1 & 1 &\cdots & 1\\ 0 & 2 & ... &1\\ \vdots&\vdots&\ddots&\vdots&\\ 0 & 0&\cdots& n & \end{pmatrix} \cdot \begin{pmatrix} x_1 \\ x_2 \\ \vdots\\ x_n \end{pmatrix} = \begin{pmatrix} \frac{1}{n} \\ \frac{2}{n} \\ \vdots\\ 1 \end{pmatrix}$$
If I solve this linear equation from bottom to top, so starting from $x_n$, I do get the solution like above. But only for an explicit row and I was not able to obtain the the solution for an arbitary $i \in \{1,\cdots, n\}$.
So, my next idea was to do it by induction over the matrix. I did the induction step for a $2\times2$-matrix. For the induction step, I assume the above solution is true for an $n \times n$-matrix.And now I like to prove it is still true for the $(n+1) \times (n+1) $-matrix
But I cannot figure out how to do the induction step from there on. Do you have any idea for the induction step or do you think induction is really the right approach here and if not, what other ways are there to prove this?