Question

If the sum of first p terms, first q terms and first r terms of an A.P.be x,y and z respectively, then x/p(q-r)+y/q(r-p)+z/r(p-q) is​

Answer 1

Step-by-step explanation:

Let the first term is A and common difference is d .

Sp = a

P/2{2A + (P-1)d} = a-----(1)

Sq = b

q/2{2A + (q-1)d} = b------(2)

Sr = c

r/2{2A + (r -1)d} = c------(3)

Now,

LHS = a(q-r)/p + b(r-p)/q + c(p-q)/r

= (q-r)/p ×P/2{2A + (p-1)d} + (r-p)/q ×q/2{2A + (q -1)d} + (p-q)/r × r/2{2A + (r-1)d}

= 1/2 [(q-r)(2A +pd -d) + (r -p)(2A + qd - d)+(p-q)(2A + rd-d)]

= 1/2[2A{(q-r)+(r-p)+(p-q)} +d{p(q-r)+q(r-p)+r(p-q)-d{(q-r)+(r-p)+(p-q)}]

= 1/2[2A{0} +d{pq-pr+qr-qp+rp-rq}-d{0}]

=1/2[0 + 0 + 0]

= 0 = RHS