A Geometric Progression (G.P.)

A side of an equilateral triangle is $20 \mathrm{cm}$ long. A second equilateral triangle is inscribed in it by joining the midpoints of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

The sum or difference of two G.P.s, is again a G.P.

The third term of a G.P is $4$, then the product of the first five terms is _____

The lengths of three unequal edges of a rectangular solid block are in G.P. The volume of the block is $216 {\mathrm{cm}}^3$ and the total surface area is $252 {\mathrm{cm}}^2$. The length of the longest edge is

The third term of $\mathrm{G}.\mathrm{P}$ is $4$, then product of its first $5$ terms is

If the $p^{\mathrm{th}}$ and $q^{\mathrm{th}}$ terms of a G.P. are $q$ and $p$ respectively, then show that its $(p+q{)}^{\mathrm{th}}$ term is ${\left(\frac{q^p}{p^q}\right)}^{\frac{1}{p-q}}$.