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

As  a,b,c are in A.P. 

therefore , 2b = a + c 

And a , b and d are in G.P. 

therefore b^2 = ad 

b^2  - ac = ad - ac

a ^ 2 + b ^ 2 -a^2 - ac = ad - ac

a ^ 2  + b ^ 2 - a (a+c) = ad - ac

a ^ 2 + b ^ 2 - 2ab = a (d -c)

(a - b) ^ 2 = a (d-c)

Thus , a ,a-b , d-c are in G.P .

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