If `a_1,a_2,a_3,...,a_n in RR ` and all are non-negative, how can I prove this? `(sqrt(a_1a_2)+sqrt(a_1a_3)+....+sqrt(a_1a_n))+` `(sqrt(a_2a_3)+...+sqrt(a_2a_n))+` `...`...

If `a_1,a_2,a_3,...,a_n in RR ` and all are non-negative, how can I prove this?






Asked on by daliasimon

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted on

I tried to edit your original attempt at writing the inequality by splitting it up over several lines. It's still a little awkward looking, but I think it's still pretty readable. I also made the assumtion that all the numbers are non-negative, since that turns out to be necessary. For the proof, we need the AM-GM inequality, which states that for any non-negative `a_j,a_k` we have `sqrt(a_ja_k)<=(a_j+a_k)/2.` Using this for every square root that occurs, we see that the left hand side of the inequality is less than or equal to






Here there are `n-1` occurrences of `a_1/2,` all in the first line. There are also `n-1` occurrences of `a_2/2,` one in the first line and `n-2` in the second line. This pattern continues, and there are `n-1` occurrences of each number `a_j/2.` Thus the intimidating sum above simplifies to `(n-1)((a_1)/2+(a_2)/2+...+(a_n)/2),` which is what we needed to show.

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daliasimon | eNotes Newbie

Posted on

It didn't let me write the entire exercise:


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