If a, b, c are in

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.