The author proves almost wordlessly the Arithmetic Mean / Geometric Mean Inequality

Mandal, Wasim Akram

OSU Libraries Digital Journal Publishing (Ohio State University)

Proof Without Words:Arithmetic Mean / GeometricMean InequalityWasim Akram Mandal, Beldanga D.H.Sr. MadrasahIn this proof without words, we prove almost wordlessly the following inequality, if c1, c2 ≥ 0, thenc1+c22 ≥√c1c2 (AM–GM inequality).Case 1: c2 < c1y =√c1ty =√c2tm2m1c1√c1c2c2c2 c1m1 > m2 (1)⇒ c1 −√c1c2c1 − c2 >√c1c2 − c2c1 − c2 (2)⇒ c1 −√c1c2 > √c1c2 − c2 (3)⇒ c1 + c2 > 2√c1c2 (4)⇒ c1 + c22>√c1c2 (5)Case 2: c2 = c1Note that c2 = c1 =⇒ c1+c22 =√c1c2.Considering Case 1 and Case 2 together, we conclude that c1+c22 ≥√c1c2.Ohio Journal of School Mathematics 80 Page 51ReferencesLove, B. J. (1977). Proof without words: Cubes and squares.Mathematics Magazine, 50(2), 74.Nelsen R. (1993). Proofs without Words: Exercise in Visual Thinking. Washington, D.C.: MathematicalAssociation of America.Nelsen, R. (2000). Proofs without Words II: More Exercise in Visual Thinking. Washington, D.C.: Mathe-matical Association of America.Wasim AkramMandal, wasim0018@gmail.com, is an assistant teacherof Beldanga D.H.Sr. Madrasah, Murshidabad, West Bengal, India. Mr.Mandal has completed research at the University of Kalyani. His re-search interests include fuzzy inventory models, supply chain manage-ment, and applied mathematics.Page 52 Ohio Journal of School Mathematics 80

Proof Without Words: Arithmetic Mean / Geometric Mean Inequality

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Proof Without Words: Arithmetic Mean / Geometric Mean Inequality

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