Let the area of the first figure below, the solid equilateral triangle, be 1sq.unit. Then suppose that smaller equilateral triangles are successively removed step in the manner depicted below. The perimeter of the shaded region of the 2006th figure is (in sq.units)

2^nd picture:- 

side of each triangle $=\frac{1}{3^{\frac{1}{4}}}$

[$\therefore$area of 1^st picture $=1$sq unit ]

$\left[\Rightarrow \frac{\sqrt{3}}{4}{a}^2=1\Rightarrow a=\frac{2}{3^{\frac{1}{4}}}\right]$

Perimeter of 2^nd picture $=3\left(\frac{{\displaystyle 1}}{{\displaystyle 3^{\frac{{\displaystyle 1}}{{\displaystyle 4}}}}}\right)\times 3$

$\begin{array}{l}=3^{1-\frac{1}{4}}\times 3=3^{\frac{3}{4}}\times 3\\ =2\times 3^{\frac{3}{4}}\times \left(\frac{3}{2}\right)\end{array}$

3^rd picture:-

side of each triangle $=\frac{1}{{2.3}^{\frac{1}{4}}}$

perimeter $=3\times \frac{1}{2\times 3^{\frac{1}{4}}}\times 9=2\times 3^{\frac{3}{4}}\times {\left(\frac{3}{2}\right)}^2$

similarly, perimeter of 2006^th figure $=2\times 3^{\frac{3}{4}}\times {\left(\frac{3}{2}\right)}^{2005}$