I am trying to find a tricky way to proof these:
If $a,b,c,d$ are in continued proportion,prove that $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2 ,$ This result could be extended to $(a^2+b^2+c^2+d^2)(b^2+c^2+d^2+e^2)=(ab+bc+cd+de)^2$ when $a,b,c,d,e$ are in continued proportion.
The standard way for solving them could be putting $\frac{a}{b}=\frac{c}{d}=k$ then followed by substitution and which is followed by tedious algebraic manipulations,but that is not what I am looking for could these be solved in a less easy way using some other algebraic method/tricks? Please explain.