If pth,qth,rth terms of an A.P. are a,b,c respectively,prove that a(q-r)+b(r-p)+c(p-q)=0.

Let common difference of series be 'd',

So, Let first term be A,

So p^th term of the series = A + (p-1)d

or, a = A + (p-1)d........(1)

Also, q^th term of the series = A + (q-1)d

or, b = A + (q-1)d....... (2) 

Similarly, r^th term of the series = A + (r-1)d

or, c = A + (r-1)d....... (3)

Now, 

L.H.S. = a(q-r) + b(r-p) +c(p-q)

= (A+pd-d)(q-r) + (A+qd-d)(r-p) + (A+rd-d)(p-q)

= Aq - Ar + pqd - pdr - dq + dr + Ar - Ap + qrd - pqd - dr + dp + Ap - Aq + pdr - qrd -dp + dq

= Aq - Ar + pqd - pdr - dq + dr + Ar - Ap + qrd - pqd - dr + dp + Ap - Aq + pdr - qrd -dp + dq

= o = R.H.S.

        Proved