Question

If a, b, c, d are in G.P, prove that $(a^n+ b^n), (b^n+c^n), (c^n+d^n)$dn) are in G.P.

Answer 1

It is given that a, b, c,and d are in G.P.
∴ b 2 = ac … (1)
c 2 = bd … (2)
ad = bc … (3)
It has to be proved that (a n + b n ), (b n + c n ), (c n + d n ) are in G.P. i.e.,
(b n + c n ) 2 = (a n + b n ) (c n + d n )
Consider L.H.S.
(b n + c n ) 2 = b 2n + 2b n c n + c 2n
= (b 2 ) n+ 2b n c n + (c 2 ) n
= (ac) n + 2b n c n + (bd) n [Using (1) and (2)]
= a n c n + b n c n+ b n c n + b n d n
= a n c n + b n c n+ a n d n + b n d n [Using (3)]
= c n (a n + b n ) + d n (a n + b n )
= (a n + b n ) (c n + d n )
= R.H.S.
∴ (b n + c n ) 2 = (a n + b n ) (c n + d n )
Thus, (a n + b n ), (b n + c n ), and (c n + d n ) are in G.P.