Question

If pqr=1, then prove that [ 1/(1+p+q-1)] + [1/(1+q+r-1)] + [1/(1+r+p-1)] = 1

Answer 1

=[ 1/(1+p+q-1)] + [1/(1+q+r-1)] + [1/(1+r+p-1)]
=[1/(p+q)]+[1/(q+r)]+[1/(r+p)]
=[r/r(p+q)+[p/p(q+r)]+[q/q(r+p)]
=[r+p+q/2(pr+qr+pq)]

Answer 2

Our first aim is to make all denominators same

1/[1 + p + q⁻¹] + 1/[1 + q + r⁻¹] + 1/[1 + r + p⁻¹]

1/[1 + p + q⁻¹] = 1/[1 + p + 1/q] = q/ [q + pq + 1] .............1

1/[1 + q + r⁻¹] = 1/[1 + q + 1/r ]

= 1/[1 + q + pqr/r ] {as pqr = 1}

= 1/[1 + q + pq] = 1/[q + pq + 1] .............2

1/[1 + r + p⁻¹] = pqr/[pqr + r + pqr/p]

= pqr/[pqr + r + qr] = pq/[pq + 1 + q] ............3

If we add (1), (2) and (3)

q[1 + pq + q] + 1/[q + pq + 1] + PQ/[pq + 1 + q]

= [1 + pq + q]/[1 + pq + q] = 1

Hence it is proved.

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