Let a , b , c be in A.P. Let p , p ′ be the arithmetic and geometric means of a , b , and q , q ′ those of b , c (with the geometric means defined). Which identity holds?

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p = ( a + b ) /2 , q = ( b + c ) /2 , p ′2 = ab , q ′2 = b c . Hence p 2 − q 2 = ( a − c ) ( a + c + 2 b ) /4 = b ( a − c ) = ab − b c = p ′2 − q ′2 , using a + c = 2 b .

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