If ${G_1}$ and ${G_2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$is

Let $\mathrm{ABC}$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $\mathrm{ABC}$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then:

The number which should be added to the numbers $2, 14, 62$ so that the resulting numbers may be in $G.P.$, is

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

The numbers $(\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)$ will be in

The number of natural number $n$ in the interval $[1005, 2010]$ for which the polynomial. $1+x+x^2+x^3+\ldots+x^{n-1}$ divides the polynomial $1+x^2+x^4+x^6+\ldots+x^{2010}$ is