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

Explanation

We know that `a,ar,ar^2,ar^3,.....` are in G.P. with first term `a` and common ratio `r`

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

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

We need to show that `(a^n+b^n),(b^n+c^n),(c^n+a^n)`are in G.P

i.e. to show common ratio are same

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


Taking L.H.S.

`(b^n+c^n)/(a^n+b^n)`

Putting value of `b` and `c`

` " " " " " =((ar)^n+(ar^2)^n)/(a^n+(ar)^n)`

` " " " " " =(a^nr^n+a^nr^2n)/(a^n+a^nr^n)`

` " " " " " =(a^nr^n(1+r^n))/(a^n(1+r^n))`

` " " " " " =r^n`


Taking R.H.S.

`(c^n+d^n)/(b^n+c^n)`

Putting value of `b`, `d` and `c`

` " " " " " =((ar^2)^n+(ar^3)^n)/((ar)^n+(ar^2)^n)`

` " " " " " =(a^nr^(2n)+a^nr^(3n))/(a^nr^n+a^nr^(2n))`

` " " " " " =(a^n(r^(2n)+r^(3n)))/(a^n(r^n+r^(2n)))`

`(r^(2n)+r^(3n))/(r^n+r^(2n))`

` " " " " " = (r^(2n)(1+r))/(r^n(1+r))`

` " " " " " = r^n`

` " " " " " =` L.H.S.

Thus, L.H.S. = R.H.S.

Hence , `(a^n+b^n),(b^n+c^n),(c^n+a^n)`are in G.P.

Hence Proved