Login/Sign Up
Show that ${x^2} + xy + {y^2},{z^2} + zx + {x^2}and{y^2} + yz + {z^2}$ are in consecutive terms of an A.P., if x, y and z are in A.P.
Show that ${x^2} + xy + {y^2},{z^2} + zx + {x^2}and{y^2} + yz + {z^2}$ are in consecutive terms of an A.P., if x, y and z are in A.P.
Arithmetic Progressions | CBSE | Class 11 | Exercise 19.5 | Maths | RD Sharma
Show that ${x^2} + xy + {y^2},{z^2} + zx + {x^2}and{y^2} + yz + {z^2}$ are in consecutive terms of an A.P., if x, y and z are in A.P.

Answer:

Given,

x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are in AP

(z2 + zx + x2) – (x2 + xy + y2) = (y2 + yz + z2) – (z2 + zx + x2)

d = common difference,

Y = x + d and x = x + 2d

Consider the LHS,

(z2 + zx + x2) – (x2 + xy + y2)

z2 + zx – xy – y2

(x + 2d)2 + (x + 2d)x – x(x + d) – (x + d)2

x2 + 4xd + 4d2 + x2 + 2xd – x2 – xd – x2 – 2xd – d2

3xd + 3d2

Consider RHS,

(y2 + yz + z2) – (z2 + zx + x2)

y2 + yz – zx – x2

(x + d)2 + (x + d)(x + 2d) – (x + 2d)x – x2

x2 + 2dx + d2 + x2 + 2dx + xd + 2d2 – x2 – 2dx – x2

3xd + 3d2

LHS = RHS

∴ x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are in consecutive terms of A.P

Thus, proved.

i

More Material