If `a ,\ b ,\ c ,\ d` are in G.P., show that: `(a b+b c+c d)^2=(a^2+b^2+c^2)(b^2+c^2+d^2)`

Explanation

Given, `a ,\ b ,\ c ,\ d`
are in G.P.


`b=ar, c=ar^2,d=ar^3`


To prove: `(ab+bc+cd)^2=(a^2+b^2+c^2)(b^2+c^2+d^2)`



`LHS=(a^2+b^2+c^2)(b^2+c^2+d^2)`


`=(a^2+(ar)^2+(ar^2)^2)((ar)^2+(ar^2)^2+(ar^3)^2)`


`=a^2(1+r^2+r^4)a^2(r^2+r^4+r^6)`


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



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