If a, b, c ,d be in G.P. , show that
(i) `(b -c)^(2) + (c - a)^(2) +(d -b)^(2) = (a - d)^(2)`
(ii) `a^(2) + b^(2) + c^(2) , ab + bc + cd , b^(2) + c^(2) + d^(2) ` are in G.P.

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."`