Mathematicians have just created three new operation symbols, @, $,and _, each doing a different thing to a pair of numbers (All of these symbols can be derived from already-known operations and notations). The following equations using the new symbols are correct:

5@5= 2/5

2@2= 1

2@3= 5/9

3@4= 7/16

2@4= 3/8

1$0= undefined

1$4= 1/10

4$1= 1/10

10$10= 1/50

(1@1) $ 1= (√2)/10

2_2= 6

2_3= 9

6_2= 42

3_5= 30

5_6= 90

If the previous equations are correct, compute the following equation.

1 @ (1 $ (1_1))=

2_2= 6

2_3= 9

6_2= 42

3_5= 30

5_6= 90

6/2= 3 = 2+1

9/3= 3 = 2+1

42/2= 21 = 6+5+4+3+2+1

30/5= 6 = 3+2+1

90/6= 15 = 5+4+3+2+1

a_b = b*a*(a+1)/2