research problem

A Sierpinski Triangle is designed using equilateral triangles. The process involves removing smaller triangles from larger ones by joining the midpoints of the sides of the larger triangles.<-- STAGE 1

More triangles will be removed up to the nth stage. Find a pattern in the total number of the triangles removed.

STAGE NO ( y ) --------------------NO OF TRIANGLES REMOVED ( T )

1 ----------------------------------------------------------.1

2 ----------------------------------------------------------.4

3 ---------------------------------------------------------.13

4 ---------------- ----------------------------------------.40

5 -------------------------------------------------------..-121

6 --------------------------------------------------------.-364

7 ----------------------------------------------------------1093

8 ----------------------------------------------------------3280

9 ----------------------------------------------------------9841

10 --------------------------------------------------------29524

Formula

---- T = n + 3 ( x )

--------where n = 1

----------------..y = stage number

----------------..x = no. of triangles removed preceeding y

----------------..T = no. of triangles removed

Example:

---- stage 1: T = n + 3( x )

----------------= 1 + 3 ( 0 )

----------------= 1

---- stage 2: T = n + 3( x )

----------------= 1 + 3 ( 1 )

----------------= 1 + 3

----------------= 4

---- stage 10: T = n + 3( x )
----------------= 1 + 3 ( 9841 )
----------------= 1 + 29523
----------------= 29524