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If a, b, c, d are in G.P, prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.

17. If a, b, c, d are in G.P, prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.

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Given: a, b, c, d are in G.P.

To prove:(a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.

Then we can write,

     b^2=ac...............................1

    c^2=bd...............................2

   ad=bc...............................3

 

Let (a^n + b^n), (b^n + c^n), (c^n + d^n) be in GP

(b^n + c^n)^2 =(a^n + b^n) (c^n + d^n)

LHS:  (b^n + c^n)^2

(b^n + c^n)^2 =b^{2n}+c^{2n}+2b^nc^n

 (b^n + c^n)^2 =(b^2)^n+(c^2)^n+2b^nc^n                 

                      =(ac)^n+(bd)^n+2b^nc^n

                      =a^nc^n+b^nc^n+a^nd^n+b^nd^n

                       =c^n(a^n+b^n)+d^n(a^n+b^n)

                       =(a^n+b^n)(c^n+d^n)=RHS

Hence  proved 

Thus,(a^n + b^n), (b^n + c^n), (c^n + d^n) are in GP

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