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Question

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

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Solution

It is given that 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.

Since a,b,c are in A.P.,

ba=cb 2b=a+c b= a+c 2

Also, b,c,d are in G.P.

c 2 =bd d= c 2 b

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

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

Substitute value of b and d in (1),

2b c 2 = 1 c + 1 e 2( a+c ) 2 c 2 = 1 c + 1 e a+c c 2 = c+e ce a+c c = e+c e

Further simplify above equation.

( a+c )e=( e+c )c ae+ce=ec+ c 2 c 2 =ae

Thus, a,c,e are in G.P.


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