Look at the number of matches first:
The first triangle used 3 pieces.
The second triangle used 9 pieces.
The third triangle is 18.
Then you can sum up:
First = zeroth +3=0+3=3.
Second time = first time +6=3+6=9
Third time = second time +9=9+9= 18
Then you can get:
The fourth = the third+12 =18+12 = 30.
Fifth = fourth+15=30+ 15=45.
Sixth = fifth+18=45+ 18=63.
Seventh time = sixth time +2 1=63+2 1=84.
According to the law obtained above, we can sum up the following formula:
N =3N(N+ 1)/2
Then the third is equal to
3*3*(3+ 1)/2=9*2= 18
Then the fifth is equal to
3*5*(5+ 1)/2= 15*3=45
Then the sixth is equal to
3*6*(6+ 1)/2= 18*3.5=63
Then one tenth is equal to
3* 10*( 10+ 1)/2=30+5.5= 165
Look at the number of triangles:
The first one consists of 1 triangles.
The second one forms three triangles.
The third one forms six triangles.
So:
The fourth composition 10 triangles.
The fifth component 15 triangles.
The nth one constitutes N(N+ 1)/2 triangles.
The tenth one consists of10 * (10+1)/2 = 55 triangles.