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Question

If a,b,c are in A.P.; b,c,d are in G.P. and 1c,1d,1e are in A.P. prove that a,c,e are in G.P.

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Solution

Given that a,b,c are in A.P.
ba=cb
b=a+c2 ...(1)

Also, given that b,c,d are in G.P.
c2=bd ...(2)

Also, given 1c,1d,1e are in A.P.

1d1c=1e1d

2d=1c+1e ...(3)

We have to prove a,c,e are in G.P.
i.e. c2=ae
Substituting the value of d from eqn (2) in eqn (3), we get
2bc2=1c+1e

2(a+c)c2=1c+1e (by (1))
a+cc2=e+cce
a+cc=e+ce
(a+c)e=(e+c)e
ae+ce=ec+c2
c2=ae
Hence, a,c,e are in G.P

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