Find all integers n(n&gt;0) such that the which quadratic equationan+1x2−2x∑k=1n+1ak2+∑k=1nak=0has real roots for every choice of real numbers a1,a2,⋯,an+1.

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Find all integers n(n&amp;gt;0) such that the which quadratic equationan+1x2−2x∑k=1n+1ak2+∑k=1nak=0has real roots for every choice of real numbers a1,a2,⋯,an+1.

Marcos Boyer · Accepted Answer

Step 1Hint: The equation has real roots if and only if the discriminant is non-negative. This means∑k=1n+1ak2≥an+1∑k=1nak.Hint: Since this has to hold true for every choice of real numbers, if we view this as a quadratic in an+1, what conclusion can we draw about the determinant?We have an+12−an+1∑k=1nak+∑k=1nak2≥0 for all an+1, so the discriminant must be non-negative, namely (∑k=1nak)2≤4∑k=1nak2. Step 2Hence, by Cauchy Schwards, we must have n≤4.Ideally, for n≥5, you should provide an explicit counter example. (This is left as an exercise to the reader).

Find all integers \(\displaystyle{n}{\left({n}{>}{0}\right)}\) such that the

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