$\sum\limits_{r = 1}^{n – 1} {r = 1 + 2 + 3 + … + \left( {n – 1} \right)}  = \frac{{n\left( {n – 1} \right)}}{2}$

$\sum\limits_{r = 1}^{n – 1} {\left( {2r – 1} \right) = 1 + 3 + 5} $

                            $ + … + \left[ {2\left( {n – 1} \right) – 2} \right] = {\left( {n – 1} \right)^2}$

$\sum\limits_{r = 1}^{n – 1} {\left( {3r – 2} \right)}  = 1 + 4 + 7 + .. + \left( {3n – 3 – 2} \right)$

                           $ = \frac{{\left( {n – 1} \right)\left( {3n – 4} \right)}}{2}$

$\therefore \sum\limits_{r = 1}^{n – 1} {{\Delta _r}} $

$ = \begin{array}{*{20}{c}}
{\sum r }&{\sum {\left( {2r – 1} \right)} }&{\sum {\left( {3r – 2} \right)} }\\
{\frac{n}{2}}&{n – 1}&a\\
{\frac{{n\left( {n – 1} \right)}}{2}}&{{{\left( {n – 1} \right)}^2}}&{\frac{{\left( {n – 1} \right)\left( {3n – 4} \right)}}{2}}
\end{array}$

$\sum\limits_{r = 1}^{n – 1} {{\Delta _r}} $ consists of $(n-1)$  determinats in $L.H.S.$ and in $R.H.S.$ every constituents of frist row consists of $(n-1)$ elements and hence it can be splitted into sum of $(n-1)$ determinats.

$\therefore \sum\limits_{r = 1}^{n – 1} {{\Delta _r}} $

$ = \begin{array}{*{20}{c}}
{\frac{{n\left( {n – 1} \right)}}{2}}&{{{\left( {n – 1} \right)}^2}}&{\frac{{\left( {n – 1} \right)\left( {3n – 4} \right)}}{2}}\\
{\frac{n}{2}}&{n – 1}&a\\
{\frac{{n\left( {n – 1} \right)}}{2}}&{{{\left( {n – 1} \right)}^2}}&{\frac{{\left( {n – 1} \right)\left( {3n – 4} \right)}}{2}}
\end{array}$

($\because $ ${R_1}$ and ${R_3}$ are identical)

Hence, value of $\sum\limits_{r = 1}^{n – 1} {{\Delta _r}} $ is independent of both $'a'$ and $'n'$.