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

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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