R D Sharma Solutions for Chapter: Arithmetic Progressions, Exercise 5: EXERCISE 19.5

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P., then show that: $\mathrm{bc}-{\mathrm{a}}^2, \mathrm{ca}-{\mathrm{b}}^2, \mathrm{ab}-{\mathrm{c}}^2$ are in A.P.

If $\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}},\frac{\mathrm{c}+\mathrm{a}}{\mathrm{b}},\frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}}$ are in A.P., prove that: $\frac{1}{\mathrm{a}},\frac{1}{\mathrm{b}},\frac{1}{\mathrm{c}}$ are in A.P.

If $\frac{\mathrm{b}+\mathrm{c}}{\mathrm{a}}, \frac{\mathrm{c}+\mathrm{a}}{\mathrm{b}}, \frac{\mathrm{a}+\mathrm{b}}{\mathrm{c}}$ are in A.P., prove that: $\mathrm{bc}, \mathrm{ca}, \mathrm{ab}$ are in A.P.

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P., prove that: ${\left(\mathrm{a}-\mathrm{c}\right)}^2=4\left(\mathrm{a}-\mathrm{b}\right)\left(\mathrm{b}-\mathrm{c}\right)$

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P., prove that: ${\mathrm{a}}^2+{\mathrm{c}}^2+4\mathrm{ac}=2\left(\mathrm{ab}+\mathrm{bc}+\mathrm{ca}\right)$

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P., prove that: ${\mathrm{a}}^3+{\mathrm{c}}^3+6\mathrm{abc}=8{\mathrm{b}}^3$

If $\mathrm{a}\left(\frac{1}{\mathrm{b}}+\frac{1}{\mathrm{c}}\right), \mathrm{b}\left(\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{a}}\right), \mathrm{c}\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{b}}\right)$ are in A.P., prove that $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P.

Show that ${\mathrm{x}}^2+\mathrm{xy}+{\mathrm{y}}^2, {\mathrm{z}}^2+\mathrm{zx}+{\mathrm{x}}^2$ and ${\mathrm{y}}^2+\mathrm{yz}+{\mathrm{z}}^2$ are consecutives terms of an A.P., if $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$ are in A.P.