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If a,b,c, are in AP prove that: 1.(a-c)^2=4(a-b)(b-c)
Take RHS4(a-b)(b-c)Expand it4[ab -ac -b^2 +bc]Use 2b = a +c. Because a,b,c are in A.P.4[(a+c)(a+c)/2 -ac - (a+c)^2]= a^2 + c^2 - 2ac= (a -c)^2= LHSHence proved
To prove(a-c)^2=4(a-b)(b-c)Or{ (a-c)^2}/}(a-b)(b-c)} =4 ...... 1Given a,b,c are in A.P. assume the common difference be d than b-a= c-b= d And c-a= 2dPut values in LHS of 14d^2/d^2=4 = RHS
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