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.

Answer

N/a

Explanation

`therefore` `b = ar, c = ar^(2), d = ar^(3)`
To prove : `a^(n) + b^(n), b^(n) + c^(n) , c^(n) + d^(n)` are in G.P.
`rArr` `(b^(n) + c^(n))^(2) = (a^(n) + b^(n)) (c^(n) + d^(n))`
L.H.S. `= (b^(n) + c^(n))^(2) = (a^(n) r^(n) + a^(n) r^(2n))^(2)`
`= (a^(n) r^(n) + a^(n) r^(2n))(a^(n) r^(n) + a^(n) r^(2n))`
`= r^(n)(a^(n) + a^(n) r^(n))(a^(n) r^(n) + a^(n) r^(2n))`
`=(a^(n) + a^(n) r^(n))(a^(n) r^(2n) + a^(n) r^(3n))`
`=(a^(n) + b^(n))(c^(n) + d^(n)) = R.H.S.`
`therefore =a^(n) + b^(n), b^(n)+c^(n),c^(n) + d^(n)` are in G.P. Hence Proved.