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

Answer

N/a

Explanation

If common ratio is r, then b=ar, `c=ar^(2),d=ar^(3)`.
`LHS=(a^(2)+b^(2)+c^(2))(b^(2)+c^(2)+d^(2))`
`(a^(2)+a^(2)r^(2)+a^(2)r^(4))(a^(2)r^(2)+a^(2)r^(4)+a^(2)r^(6))`
`=(a^(2)+a^(2)r^(2)+a^(2)r^(4))*r(a^(2)r+a^(2)r^(3)+a^(2)r^(5))`
`=(a^(2)r+a^(2)r^(3)+a^(2)r^(5))(a^(2)r+a^(2)r^(3)+a^(2)r^(5))`
`=(a^(2)r+a^(2)r^(3)+a^(2)r^(5))^(2)`
`=(a*ar+ar*ar^(2)+ar^(2)*ar^(3))^(2)`
`=(ab+bc+cd)^(2)`=RHS Hence Proved.