If and are two real numbers such that , , are in Arithmetic Progression, prove that:
, , are also in A.P. for all
By definition, a,b,c are in arithmetic progression iff c-b=b-a ~ a+c=2b.
Given + =
Or, + =
But the sum of two squares can never equal a negative number, therefore there are no real solutions for and . Ergo, this theorem is vacuously true.