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 $AP$.

Find two positive numbers $a$ and $b$, whose

$AM=25$ and $GM=20$

Find two positive numbers $a$ and $b$, whose

$AM=10$ and $GM=8$

Find the $GM$ between the numbers

$5$ and $125$

The geometric mean of $1$ and $\frac{9}{16}$ is $\frac{k}{4}$.  Find the values of $k$.

Find the $GM$ between the numbers

$0.15$ and $0.0015$

Find the $GM$ between the numbers

$-8$ and $-2$

Find the $GM$ between the numbers

$-6.3$ and $-2.8$

The $GM$ between the numbers$a^{3}b \mathrm{and} a{b}^3$ is $a^k{b}^k$. Find the value of $k$.