If each term of a geometric progression ${\mathrm{a}}_1,{\mathrm{a}}_2,{\mathrm{a}}_3,\dots$ with ${\mathrm{a}}_1=\frac{1}{8}$ and ${\mathrm{a}}_2\ne {\mathrm{a}}_1$, is the arithmetic mean of the next two terms and ${\mathrm{S}}_{\mathrm{n}}={\mathrm{a}}_1+{\mathrm{a}}_2+\dots +{\mathrm{a}}_{\mathrm{n}}$, then ${\mathrm{S}}_{20}-{\mathrm{S}}_{18}$ is equal to

The value of sum $\sum _{n=1}^{13}\left(i^n+i^{n+1}\right),$ where $i=\sqrt{-1}$, equals

Let $S_1$ be the sum of areas of the squares whose sides are parallel to coordinate axes. Let $S_2$ be the sum of areas of the slanted squares as shown in the figure. Then $S_1/S_2$ is

If $m$ is the $A.M.$of two distinct real numbers $I$ and $n$ $\left(I, n>1\right)$  and $G_1, G_2 \text{and }{G}_3$ are three geometric means between $I \text{and} n$, then $G_1^4+2G_2^4+G_3^4$ equals