Find the tenth term of the sequence .
The figure shows the steps to producing a curve . It starts as the boundary of the unit square in Figure (a). In Figure (b), we remove a square quarter of the area of the square in (a). In Figure (c), we add a square quarter of the area that we removed in (b). In Figure (d), we remove a square quarter of the area of the square we added in (c). If we continue to do this indefinitely, we will get the curve . We let be the region enclosed by . By summing a suitable infinite series, find the area of region . Give your answer as a fraction.
An equilateral triangle has a side length of 14 cm, where another triangle is drawn inside of it by connecting the midpoints of its sides. More interior triangles are to be repeatedly drawn the same way as shown in the figure. Find the sum of the perimeters of the first 6 triangles drawn giving the answer to the nearest integer.