If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.
If a,b,c are in AP it follows that
a + c = 2b……..(1)
and a,x,b and b,y,c are in individual GPs which follows
x2 = ab …….(2)
y2 = bc ……..(3)
Adding eqn 2 and 3 we get,
x2 + y2 = ab + bc
= b(a + c)
= b.2b ( from eqn 1)
= 2b2
So we get x2 + y2 = 2b2 which shows that they are in AP.