If a, b, c, d are in G.P., then prove that:
`(b-c)^(2)+(c-a)^(2)+(d-b)^(2)=(a-d)^(2)`

Answer

N/a

Explanation

Let 'r' be the common ratio of the G.P.
`therefore b=ar, c=ar^(2)," and "d=ar^(3)`
`"L.H.S. = "(b-c)^(2)+(c-a)^(2)+(d-b)^(2)`
`=(ar-ar^(2))+(ar^(2)-a)^(2)+(ar^(3)-ar)^(2)`
`=a^(2)r^(2)-a^(2)r^(4)-2a^(2)r^(3)+a^(2)r^(4)+a^(2)-2a^(2)r^(2)+a^(2)r^(6)+a^(2)r^(2)-2a^(2)r^(4)`
`=-2a^(2)r^(3)+a^(3)+a^(2)r^(6)=(a-ar^(3))^(2)`
`=(a-d)^(2)`
`"R. H. S."`