If a, b, c, d are in G.P, prove that are in G.P.

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Solution

It is given that a, b, c,and d are in G.P.

∴b² = ac … (1)

c² = bd … (2)

ad = bc … (3)

It has to be proved that (aⁿ + bⁿ), (bⁿ + cⁿ), (cⁿ + dⁿ) are in G.P. i.e.,

(bⁿ + cⁿ)² = (aⁿ + bⁿ) (cⁿ + dⁿ)

Consider L.H.S.

(bⁿ + cⁿ)² = b²ⁿ+ 2bⁿcⁿ + c²ⁿ

= (b²)ⁿ+ 2bⁿcⁿ + (c²)ⁿ

= (ac)ⁿ + 2bⁿcⁿ + (bd)ⁿ [Using (1) and (2)]

= aⁿ cⁿ + bⁿcⁿ+ bⁿ cⁿ + bⁿ dⁿ

= aⁿ cⁿ + bⁿcⁿ+ aⁿ dⁿ + bⁿ dⁿ [Using (3)]

= cⁿ (aⁿ + bⁿ) + dⁿ (aⁿ + bⁿ)

= (aⁿ + bⁿ) (cⁿ + dⁿ)

= R.H.S.

∴ (bⁿ + cⁿ)² = (aⁿ + bⁿ) (cⁿ + dⁿ)

Thus, (aⁿ + bⁿ), (bⁿ + cⁿ), and (cⁿ + dⁿ) are in G.P.

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