Let a,b,c,d,p be any nonzero distinct real numbers such that (a^2+b^2+c^2)p^2 - 2(ab+bc+cd…

Question

Let $a,b,c,d,p$ be any nonzero distinct real numbers such that $(a^2+b^2+c^2)p^2 - 2(ab+bc+cd)p+(b^2+c^2+d^2)=0$. Then:

Accepted Answer

Rearranging: $(ap-b)^2+(bp-c)^2+(cp-d)^2=0$, so each term is zero:

$$ap=b,\ bp=c,\ cp=d \implies \frac ba=\frac cb=\frac dc=p.$$

So $a,b,c,d$ are in G.P.

Solutions · 1