If a,b,c,d and p are distinct real numbers such that (a^2+b^2+c^2)p^2-2p(ab+bc+cd)+(b^2+c^…

Question

If $a,b,c,d$ and $p$ are distinct real numbers such that $(a^2+b^2+c^2)p^2-2p(ab+bc+cd)+(b^2+c^2+d^2)\le0$, then:

Accepted Answer

Rewrite the expression as a sum of squares:
$$(a^2p^2-2abp+b^2)+(b^2p^2-2bcp+c^2)+(c^2p^2-2cdp+d^2)=(ap-b)^2+(bp-c)^2+(cp-d)^2\le0.$$

Since each square is non-negative, the sum is $\le0$ only if each is $0$: $ap=b,\ bp=c,\ cp=d$, i.e. $p=\dfrac ba=\dfrac cb=\dfrac dc$. Hence $a,b,c,d$ are in G.P.

Solutions · 1