arberiii wrote: It's given system of equations $a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$ $a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$ .......... $a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$ such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it Let $a+k\Delta$ be the arithmetic sequence (with $a=a_{11}$ and $\Delta=a_{12}-a_{11}$) If $a=\Delta=0$ then any $x_i$ are solutions If $a\ne 0$ and $\Delta=0$ system is just $\sum x_i=1$ If $\Delta\ne 0$ and $n=1$, the system is $ax_1=a+\Delta$ and has a unique solution if $a\ne 0$ If $\Delta\ne 0$ and $n>1$, subtracting lines, we get that system is equivalent to : $\sum x_i=1$ and $\sum_{k=2}^n(k-1)x_k=n$ From there, I need precision on your statement. Is "one" meaning "one unique" or "at least one" ? If it means "one unique solution", we have two cases : $n=1$ and $a\ne 0$ and $\Delta\ne 0$ and solution is $x_1=1+\frac{\Delta}a$ $n=2$ and $\Delta\ne 0$ and solution is $(-1,2)$

Any other solution?

In all other cases, there are infinitely many solutions.