Answer :

Proof: ∵ a, b, c, d are in G.P

⇒ a = a, b = ar ,c = ar^{2},d = ar^{3}.

To prove: a + b, b + c, c + d, are also in G.P, if-

⇒ (a + b) (c + d) = (b + c)^{2}

Now, we need to prove : (a + b) (c + d) = (b + c)^{2}

L.H.S. = (a + ar)(ar^{2} + ar^{3})

= a(1 + r) ar^{2} (1 + r)

= a^{2}r^{2} (1 + r)^{2}

R.H.S. = (ar + ar^{2})^{2}

= (ar(1 + r))^{2}

= a^{2}r^{2} (1 + r)^{2}

⇒ L.H.S = R.H.S

Hence, proved that-

(a + b) (c + d) = (b + c)^{2}

⇒ a + b, b + c, c + d, are also in G.P

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