let the positive no. a,b,c,d be in A.P. Then abc,bcd,cda,dab areA) NOT in AP/GP/HPB) in APC) in GPD) in HP

Dear Raman Malik,

a,b,c,d are positive numbers in A.P.

Note: If a=b=c=d, then the answer will be AP,GP and HP i.e., option B,C,D should be correct. But in general, we consider that they are distinct.

we have to check whether abc,bcd,cda,dab follows AP/GP/HP.

i.e., we have to check whether abcd/d, abcd/a, abcd/b, abcd/c follows AP/GP/HP. 

i.e., we have to check whether 1/d, 1/a, 1/b, 1/c follows AP/GP/HP. 

As a,b,c,d are in AP, they will be either strictly increasing or decreasing.

Hence, 1/a, 1/b, 1/c, 1/d are also either strictly increasing or strictly decreasing sequence.

=> 1/d, 1/a, 1/b, 1/c is not either strictly increasing or strictly decreasing.

=> 1/d, 1/a, 1/b, 1/c  is not in AP/GP/HP.

Hence, option A is the correct option.

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Nagesh.