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If a 2 , b 2 , c 2 are in A.P , prove that 1/b+c, 1/c+a, 1/a+b. If a2 , b2 , c2 are in A.P , prove that 1/b+c, 1/c+a, 1/a+b.
Answer :;In AP form 2nd term - 1st term = 3rd term - 2nd termb²-a² = c²-b²b²+b² = c²+a²2b² = c²+a²Add2ab+2ac+2bc on both sides2b²+2ab+2ac+2bc = a²+c²+ac+ac+bc+bc+ab+ab2b²+2ab+2ac+2bc = ac+bc+a²+ab+bc+c²+ab+ac2b²+2ab+2ca+2cb = ca+cb+a²+ab+cb+c²+ab+ac2(ba+b²+ca+cb) = (ca+cb+a²+ab) + (cb+c²+ab+ac)2((ba+b²)+(ca+cb)) = ((ca+cb)+(a²+ab)) + ((cb+c²)+(ab+ac))2(b(a+b)+c(a+b)) = (c(a+b)+a(a+b)) + (c(b+c)+a(b+c))2(b+c)(a+b) = (c+a)(a+b) + (c+a)(b+c)Divide whole by (a+b)(b+c)(c+a),we get,2/c+a = 1/b+c + 1/a+bhence proved.Thanks
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