Sums of the first p,q,r terms …

Let A be the first term and D be the common difference of the given A.P. Then,
a = Sum of p terms
{tex}\Rightarrow \quad a = \frac { p } { 2 } \{ 2 A + ( p - 1 ) D \}{/tex}
{tex}\Rightarrow \quad \frac { 2 a } { p } = \{ 2 A + ( p - 1 ) D \}{/tex}...(i)
b = Sum of q terms
{tex}\Rightarrow \quad b = \frac { q } { 2 } \{ 2 A + ( q - 1 ) D \}{/tex}
{tex}\Rightarrow \quad \frac { 2 b } { q } = \{ 2 A + ( q - 1 ) D \}{/tex}...(ii)
and, c = Sum of r terms
{tex}\Rightarrow \quad c = \frac { r } { 2 } \{ 2 A + ( r - 1 ) D \}{/tex}
{tex}\Rightarrow \quad \frac { 2 c } { r } = \{ 2 A + ( r - 1 ) D \}{/tex}...(iii)
Multiplying equation (i), (ii) and (iii) by (q - r), (r - p) and (p - q) respectively and adding, we get
{tex}\frac { 2 a } { p } ( q - r ) + \frac { 2 b } { q } ( r - p ) + \frac { 2 c } { r } ( p - q ){/tex}
= {2A + (p - 1)D} (q - r) + {2A + (q - 1)D} (r - p)+ {2A+(r - 1)D} (p-q)
= 2Aq - 2Ar + (p-1)(q -r) D + 2Ar - 2AP + (q - 1) (r -p)D + 2Ap - 2Aq + (r -1)(p-q)A
= 2Aq - 2Ar + 2Ar - 2AP+ 2Ap - 2Aq + (p-1)(q -r) D + (q - 1) (r -p)D + (r -1)(p-q)A
= 2A (q - r + r - p + p - q) + D {(p-1) (q - r) + (q - 1) (r - p) + (r - 1) (p-q)}
= 2A × 0 + D × 0
= 0