Discrete Mathematics

r =   -80/120  = -2/3

Sum of 1st  8 terms  

S₈  =  first term  *  [ 1 - r^8 ] / [ 1  - r ]  =

120*  [  1 - (-2/3)^8 ] / [ 1 - (-2/3) ]  =   50440 /   729 

OK

r = - 80 / 120  =  -2/3

Sum of 1st n terms of a geometric series = first term * [ 1 - r^n ] / [1 - r ]   =

120 *  [ 1 - (-2/3)^8 ] /  (1 - (-2/3) )

Taking it  by parts

[1 - (-2/3)^8  ] =    

[1 - 256/6561 ]  =

[ 6561 - 256 ] / 6561    =

[6305 / 6561 ]

And  ( 1 - -2/3 )  = ( 1 + 2/3)  = (5/3)

So we have

120 / ( 5/3)  *  ( 6305 /6561)  =

72 * (6305 / 6561)   =  453960 /  6561  =    [50440 * 9  ]  /  [ 729 * 9 ]  =  50440 / 729