Answer:

	Going Down
Triangle( 4 ) =	+ Triangle( )	= +	=
Triangle( 3 ) =	+ Triangle( )	= +	=
Triangle( 2 ) =	+ Triangle( )	= +	=
Triangle( 1 ) =

		Going Up
Triangle( 4 ) =	+ Triangle( )	= +	=
Triangle( 3 ) =	+ Triangle( )	= +	=
Triangle( 2 ) =	+ Triangle( )	= +	=
Triangle( 1 ) =

Example

Here is the formula for triangle numbers:

Triangle( 1 ) = 1
Triangle( N ) = N + Triangle( N-1 )

Here is an example that calculates Triangle( 5 ):

Triangle( 5 ) = 5 +  Triangle( 4 )

              = 5 + ( 4 + Triangle( 3 ) )

              = 5 + ( 4 + ( 3 + Triangle( 2 ) ))

              = 5 + ( 4 + ( 3 + ( 2 + Triangle( 1 ) )))
              
              = 5 + ( 4 + ( 3 + ( 2 +      1        )))
              
              = 5 + ( 4 + ( 3 + 3) )
              
              = 5 + ( 4 + 6 )
              
              = 5 + 10
              
              = 15

QUESTION 7:

Can Triangle(N) be calculated for all positive integers N?