Show that The following series is in GP ( Geometric Progression ) ?

(a^2 + b^2+ c^2) , (ab+bc+cd),(b^2+c^2+d^2)(a2+b2+c2),(ab+bc+cd),(b2+c2+d2)
Prove It is in Geometric Progression

1 Answer
Aug 8, 2018

Please refer to The Explanation.

Explanation:

I think the Problem is to prove :

If a,b,c,da,b,c,d are in GP, then, show that,

(a^2+b^2+c^2), (ab+bc+cd), (b^2+c^2+d^2)(a2+b2+c2),(ab+bc+cd),(b2+c2+d2) are also in GP.

It suffices to show that,

(ab+bc+cd)^2=(a^2+b^2+c^2)(b^2+c^2+d^2)..........(star).

Given that, a,b,c,d are in GP.

:. b/a=c/b=d/c=r," say".

:. b=ar, c=br=(ar)r=ar^2, and, d=cr=ar^3...(star').

Utilising (star') in the R.H.S. of (star), we have,

"The R.H.S. of "(star)=(a^2+b^2+c^2)(b^2+c^2+d^2),

=(a^2+a^2r^2+a^2r^4)(a^2r^2+a^2r^4+a^2r^6),

=a^2(1+r^2+r^4){a^2r^2(1+r^2+r^4)},

=a^4r^2(1+r^2+r^4)^2,

={a^2r(1+r^2+r^4)}^2,

=(a^2r+a^2r^3+a^2r^5)^2,

=(a*ar+ar*ar^2+ar^2*ar^3)^2,

=(ab+bc+cd)^2,

"=The L.H.S. of "(star).

Hence, the Proof.