Question

if a,b,c,d are in G.P.,than show that ab,ca,bc,bd,,re also in G.P.

Answer 1

a/b=b/c=c/d=k
then we get
a=bk
b=ck
c=dk
now.,
ab/ca=ca/bc=bc/bd
b/c=a/b=c/d
ck/c=bk/b=dk/d
k=k=k
so they are in GP

Answer 2

a , b , c , d are in GP
Assume a = P , b = Pr , c = Pr² and d = Pr³
[ Here , a , b, c , d are in GP so, we have to assume a , b, c and d in such a way that common ratio of series must be same ]
Now,
a = P
b = Pr
c = Pr²
d = Pr³

So, ab = P × Pr = P²r
ca = Pr² × P = P²r²
bc = Pr × Pr² = P²r³
bd = Pr × Pr³ = P²r⁴

Here we see , P²r , P²r², P²r³, P²r⁴ are in GP where common ratio is r and first term is P²r
Hence, ab , ca, bc , bd are in GP