Question

If pqr=1, then prove that [ 1/(1+p+q-1)] + [1/(1+q+r-1)] + [1/(1+r+p-1)] = 1. ....... plzz no google answers

Answer 1

LHS = 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 ] {from 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)

Add equations (1), (2) and (3)
q[1 + pq + q] + 1/[q + pq + 1] + PQ/[pq + 1 + q]
= [1 + pq + q]/[1 + pq + q] = 1
Hence , LHS = RHS proved//

Answer 2

Given :

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

PROVED !