If `a ,\ b ,\ c` are in A.P. `b ,\ c ,\ d` are in G.P. and `1/c ,1/d ,1/e` are in A.P. prove that `a ,\ c ,\ e` are in G.P.?

Options

A. a,c,e are in G.P.
B. a,b,e are in G.P.
C. a,b,e are in G.P.
D. a,c,e are in G.P.

Answer

A

Explanation

It si given that :
a,b,c are in A.P. `rArr2b=a+c` . . .(i)
b,c,d are in G.P. `rArrc^(2)=bd` . . .(ii)
`(1)/(c),(1)/(d),(1)/(e)` are in A.P. `rArr(2)/(d)=(1)/(c)+(1)/(e)rArrd=(2ce)/(c+e)` . . .(iii)
From (ii) and (iii), we obtain
`c^(2)=b((2ce)/(c+e))" [On eliminating d]"`
`rArr" "2be=c^(2)+ce`
`rArr" "(a+c)e=c^(2)+ce`
`rArr" "c^(2)=ae" [Using (i)]`
`rArr" "a,c,e,` are in G.P.