Assume p, q and r are distinct positive integers. Suppose the p th, q th, r th terms of a nonconstant A.P. form a G.P. in that order. What is the G.P.'s common ratio, assuming it is defined?

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Let the selected terms be u , u R , u R 2 and the A.P. common difference be d  = 0 . Then ( q − p ) d = u ( R − 1 ) and ( r − q ) d = u R ( R − 1 ) . Since d  = 0 , R  = 1 and division yields R = ( r − q ) / ( q − p ) = ( q − r ) / ( p − q ) .

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