If a, b, c , d are in G.P. , then shown that
(i) `(a + b)^(2) , (b +c)^(2), (c + d)^(2)` are in G.P.
(ii) ` (1)/(a^(2) + b^(2)), (1)/(b^(2) +c^(2)), (1)/(c^(2) + a^(2))` are in G.P.

Explanation

(i) Given, a, b,c , d are in G.P.
Let r be the common ratio of the G.P.
then ` b = ar , c = ar^(2) , d= ar^(3)`
Now , `((b+c)^(2))/((a + b)^(2)) = ((ar + ar^(2))^(2))/((a + ar)^(2)) = (a^(2)r^(2) (1 + r)^(2))/(a^(2) (1 + r)^(2)) = r^(2)`
and `((c+d)^(2))/(( b + c)^(2)) = ((ar^(2) + ar^(3))^(2))/((ar + ar^(2))^(2)) = (a^(2)r^(4) (1 + r)^(2))/(a^(2) r^(2) (1 + r)^(2)) = r^(2)`
Clearly , ` ((b + c)^(2))/((a + b)^(2)) = ((c +d)^(2))/((b + c)^(2)) = r^(2)`
Hence ` (a + b)^(2) , (b + c)^(2) , (c + d)^(2)` are in G.P.
(ii) Let r be the common ratio .