If *a*,* b*,*
c*,* d* are in G.P, prove that
are in G.P.

It is given that *a*,* b*,* c*,and *d* are in G.P.

∴*b*^{2} = *ac* … (1)

*c*^{2} = *bd* … (2)

*ad* = *bc* … (3)

It has to be proved that (*a*^{n} + *b*^{n}), (*b*^{n} + *c*^{n}), (*c*^{n} + *d*^{n}) are in G.P. i.e.,

(*b*^{n} + *c*^{n})^{2} = (*a*^{n} + *b*^{n}) (*c*^{n} + *d*^{n})

Consider L.H.S.

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

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

= (*ac*)^{n} + 2*b*^{n}*c*^{n} + (*bd*)^{n} [Using (1) and (2)]

= *a*^{n} *c*^{n} + *b*^{n}*c*^{n}+ *b*^{n} *c*^{n} + *b*^{n} *d*^{n}

= *a*^{n} *c*^{n} + *b*^{n}*c*^{n}+ *a*^{n} *d*^{n} + *b*^{n} *d*^{n} [Using (3)]

= *c*^{n} (*a*^{n} + *b*^{n}) + *d*^{n} (*a*^{n} + *b*^{n})

= (*a*^{n} + *b*^{n}) (*c*^{n} + *d*^{n})

= R.H.S.

∴ (*b*^{n} + *c*^{n})^{2} = (*a*^{n} + *b*^{n}) (*c*^{n} + *d*^{n})

Thus, (*a*^{n} + *b*^{n}), (*b*^{n} + *c*^{n}), and (*c*^{n} + *d*^{n}) are in G.P.

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