R S Aggarwal Solutions for Exercise 5: EXERCISE 12E

If $a, b, c$ are in $\mathrm{GP},$ prove that $a^2,b^2,c^2$ are in $\mathrm{GP}$.

If $a, b, c$ are in $GP,$ prove that $a^3,b^3,c^3$ are in $GP$

If $a, b, c$ are in $GP,$ prove that $\left(a^2+b^2\right),(ab+bc),\left(b^2+c^2\right)$ are in $GP$.

If $a, b, c, d$ are in $GP,$ prove that $\left(a^2-b^2\right),\left(b^2-c^2\right),\left(c^2-d^2\right)$ are in $GP$.

If $a, b, c, d$ are in $GP,$ then prove that $\frac{1}{\left(a^2+b^2\right)},\frac{1}{\left(b^2+c^2\right)},\frac{1}{\left(c^2+d^2\right)}$ are in $GP$.

If $\left(p^2+q^2\right),(pq+qr),\left(q^2+r^2\right)$ are in $GP$ then prove that $p, q, r$ are in $GP$.

If $a, b, c$ are in $AP,$ and $a, b, d$ are in $GP,$ show that $a,(a-b)$ and $(d-c)$ are in $GP$.

If $a, b, c$ are in $AP,$ and $a, x, b$ and $b, y, c$ are in $GP$ then show that $x^2,b^2,{\mathrm{y}}^2$ are in $AP$.