Find total no of triangle in the following diagram.....
It takes too much time to count. The counting will we too difficult on increasing the no of levels........
Here i am going to give a general procedure to solve such type of problems......
Lets start with level one........If there is only one triangle
then total no of triangle will be one.
T(1) = Tri(1)
= 1
Now for the level two....
The total no of triangle will be
T(2) = Tri(1) +Tri(2)
= 4 + 1
= 5
Here T(n) represent the total no of triangle and Tri(n) represents the triangles of size n.
For level 3 you can see there is 9 triangles of size 1, 3 triangles of size 2 and 1 of size 3......
T(3) = Tri(1) + Tri(2) + Tri(3)
= 9 + 3 + 1
= 13
For level 4 you can see there is 16 triangles of size 1, 7 triangles of size 2, 3 of size 3 and 1 of size 4.
T(4) = Tri(1) + Tri(2) + Tri(3) + Tri(4)
= 16 + 7 + 3 + 1
= 27
Now we have a sequence 1,5,13,27..........
put this sequence on "http://www.research.att.com/~njas/sequences/"
you will get the formula for the any sequence which exist.
I found the that the above sequence does not exist but it is approximate to
a sequence so the formula for the nth term is Floor[n*(n+2)*(2n+1)/8........
using formula for the size 5 (level 5)
T(5) = Floor(5*7*11/8)
= Floor(48.12500)
= 48
hence for the size 6 (level 6)
T(6) = Floor(6*8*13/8)
= Floor(78)
= 78
