Saturday, 22 November 2014

algebra precalculus - Prove sumlimitsni=1fraca2ibigeqfrac(sumlimitsni=1ai)2sumlimitsni=1bi




So I have the following problem, which I'm having trouble solving:



Let a1 , a2 , ... , an be real numbers. Let b1 , b2 , ... , bn be positive real numbers. Prove



a21b1+a22b2++a2nbn(a1+a2++an)2b1+b2++bn



I was thinking that I somehow could use the Cauchy–Schwarz inequality, but with no success.



Any help would be very appreciated


Answer




You can simply use the cauchy-scwartz on the sets {a1b1,a2b2,,anbn} and {b1,b2,,bn}.



What you will get, is (a21b1+a22b2++a2nbn)(b1+b2++bn)(a1+a2++an)2.


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