Prove that aq – r. br – p. cp – q = 1.

If a = xyp – 1, b = xyq – 1, and c = xyr – 1, prove that aq – r. br – p. cp – q = 1.

Solution:

It is given that

a = xyp – 1

Here

aq – r = (xyb – 1)q – r = xq – r. y(q – r) (p – 1)

b = xyq – 1

Here

br – p = (xyq – 1)r – p = xr – p. y(q – 1) (r – p)

c = xyr – 1

Here

cp – q = (xyr – 1)p – q = xp – q. y(r – 1) (p – q)

Consider

LHS = aq – r. br – p. cp – q

Substituting the values

= xq – r. y(q – r) (p – 1). xr – p. y(q – 1) (r – p). xp – q. y(r – 1) (p – q)

By further calculation

= xq – r + r – p – q. y(p – 1) (q – r) + (q – 1) (r – p) + (r – 1) (p – q)

So we get

= x0. ypq – pr – q + r + qr – pr – r + p + rp – qr – p + q

= x0. y0

= 1 × 1

= 1

= RHS

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