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

Question

#(a^2 + b^2+ c^2) , (ab+bc+cd),(b^2+c^2+d^2)#
Prove It is in Geometric Progression

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Answer 1 · 2018-08-08T09:50:27

Please refer to The Explanation.

I think the Problem is to prove :

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

#(a^2+b^2+c^2), (ab+bc+cd), (b^2+c^2+d^2)# 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.