If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that

1.	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.
Answer» 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.

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.

Answer»

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.