I have a question about close form formula of this pattern. As you see the first one has $a(1)=6$ matches,$a(2)=24$and $a(3)=48$
The question ask for $a(10)$ , I found the number of matches as below $$a(10)=6+(2*6+6)+(3*6+6)+(4*6+6)+\cdots+(10*6+6)$$ but It seems a gap between $a(1),a(2)$ to generalize it. Can someone help me? Or bring some other Idea to find $a(n)=?$
I appreciate any help you can provide. ( I do apologize for the low-quality picture)
$$a(1)=3\times 6 - 12\\ a(2)=6\times 6 - 12\\ a(3)=10\times 6 - 12\\ a(4)=15\times 6 - 12$$
Notice the triagular numbers.
$$a(n)=6\times T_{n+1}-12$$