The sum of three numbers $a, b, c$ in $\mathrm{A}.\mathrm{P}.$ is $18$. If $a$ and $b$ are each increased by $4$ and $c$ is increased by $36$, the new numbers form a $\mathrm{G}.\mathrm{P}$. Find $a, b, c$.

If a, b, c are in G.P., prove that:

$a^2{b}^2{c}^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=a^3+b^3+c^3$

If a, b, c are in G.P., prove that:

$\frac{{\left(a+b+c\right)}^2}{a^2+b^2+c^2}=\frac{a+b+c}{a-b+c}$

If a, b, c are in G.P., prove that:

$\frac{1}{a^2-b^2}+\frac{1}{b^2}=\frac{1}{b^2-c^2}$

If a, b, c are in G.P., prove that:

$\left(a+2b+2c\right)\left(a-2b+2c\right)=a^2+4c^2$

If $a,b,c,d$ are in G.P., prove that:

${\left(a+b+c+d\right)}^2={\left(a+b\right)}^2+2{\left(b+c\right)}^2+{\left(c+d\right)}^2$