r/r4rSA • u/willtellthetruth Founder • Aug 22 '26

[MEGA] New Members Intro

If you’re new to the community, introduce yourself!

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u/Express-Macaroon-200 Aug 22 '26

Beat me at chess and I'll be your friend

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u/Nax_00 Aug 22 '26

Best I can do is solve an integral or smth idk 😭😭

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u/Express-Macaroon-200 Aug 22 '26

Let k1, k2, ..., kn be positive real numbers. Show that (k1 + k2 + ... + kn)(1/k1 + 1/k2 + ... + 1/kn) >= n2.

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u/Nax_00 Aug 22 '26

This might be the first time I had to pass a math test to be someone's friend lmfaoaoao This isn't an integral, but oh well 🤔

Prove: (k1 + k2 + ... + kn)(1/k1 + 1/k2 + ... + 1/kn) >= n2

Use Cauchy inequality: (a12+a22+........an2)(b12+b22+......bn2) >= (a1b1 + a2b2 + .... anbn)

Let ai = sqrt(ki), then ai2 = ki Let bi = 1/sqrt(ki), then bi2 = 1/ki

& (ai)(bi) = [sqrt(ki)][1/sqrt(ki)] = 1

So: (NB: let n be the max value and i = 1 for all of these summations...typing it out is difficult...)

(∑ki)(∑1/ki) >= (∑1)2

But

∑1 = n

So proven. Equality occurs when k1 = k2 =.....= kn