If a, b,c, d are in G P, prove that are in G P - Maths - Sequences and Series

Question

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

Gopal Mohanty · Accepted Answer

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

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