एक `A.P.` के `p,q,r` पदों के योग क्रमशः `a,b,c` है सिद्ध कीजिए की
` " "(a)/(p) (q-r) +(b)/(q) (r-p) +(c)/(r) (p-q) =0`

Explanation

माना `A.P.` का प्रथम पद `'x'` और सार्वान्तर 'y ' है|
` therefore " " s_p=a`
` rArr" " (p)/(2) [2x+(p-1)y]=a` `rArr " " (a)/(p) =x+(p-1)(y)/(2)`
` rArr " "(a)/(p) (q-r) =x(q-r) +(p-1)(q-r) (y)/(2)`
इसी प्रकार,`" " S_q =b`
` rArr" " (q)/(2) [2x+(q-1)y]=b`
` rArr " "(b)/(q) =x+(q-1)(y)/(2)` `rArr" "(b)/(q)(r-p)=x(r-p)+(q-1)(r-p)(y)/(2)`
और `" "S_r =c`
`rArr" "(r)/(2)[2x+(r-1)y]=c``rArr" " (c)/(r) =x+(r-1)(y)/(2)`
` rArr" " (c)/(r) (p-q)=x(p-q)+(r-1)(p-q)(y)/(2) `
समीकरण (1 ) ,(2 ) और (3 ) को जोड़ने पर,
` (a)/(p) (q-r) +(b)/(q)(r-p) +(c)/(r) (p-q) =x{(q-r)+(r-p)+(-q)}+(y)/(2){(p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q)}`
` " " =0+(y)/(2){pq-pr -q+r+qr-pq-r+p+rp-rq-p+q}`
`" " =0`