# Math Help - progressions!

1. ## progressions!

if a,b,c,d and p are distinct real numbers such that
$(a^2 +b^2 +c^2 )p^2 -2(ab + bc + cd)p+(b^2 + c^2 + d^2)$is less than or equal to zero..

then prove that $a,b,c,d$ are in geometric progression.

thanks for you reply.

2. Originally Posted by grgrsanjay
if a,b,c,d and p are distinct real numbers such that
$(a^2 +b^2 +c^2 )p^2 -2(ab + bc + cd)p+(b^2 + c^2 + d^2)$is less than or equal to zero.

then prove that $a,b,c,d$ are in geometric progression.
If a quadratic (in p), in which the coefficient of $p^2$ is non-negative, can take a value less than or equal to zero, then its discriminant must be greater than or equal to zero. Therefore $(ab+bc+cd)^2 - (a^2+b^2+c^2)(b^2+c^2+d^2) \geqslant0$. Multiply that out and rearrange it as $-(b^2-ac)^2 -(c^2-bd)^2 - (ad-bc)^2 \geqslant0$. Can you take it from there?

3. a:b:c:d ; p
-2:-4:-8:-16 ; 2
2:-4:8:-16 ; -2

For your info, that pattern exists.

4. thanks! i got it