n = 8
the sum to n terms of a geometric sequence is
[tex]S_{n}[/tex] = [tex]\frac{a(1-r^{n )} }{1-r}[/tex]
where a is the first term and r the common ratio
here a = 16 and r = [tex]\frac{64}{-32}[/tex] = [tex]\frac{-32}{16}[/tex] = - 2
[tex]S_{n}[/tex] = ( 16( 1 - ( - 2 )^n )/1 - (- 2 )) = - 1360
[tex]\frac{16}{3}[/tex] ( 1 - (- 2 )^n ) = - 1360
multiply both sides by [tex]\frac{3}{16}[/tex]
1 - (- 2 )^n = - 255
- (- 2 )^n = - 256
)-2)^n = 256
note that [tex]2^{8}[/tex] and [tex](-2)^{8}[/tex] = 256
hence n = 8
