Enter a recursive rule for the geometric sequence. 10,−80,640,− a1= ; an=

A recursive rule for the geometric sequence is:
a1=10
an=-8 (an-1)

Solution:

Sequence:
10, -80, 640, -5120

a1=10
a2=-80
a3=640
a4=-5120

a2/a1=(-80)/(10)→a2/a1=-8
a3/a2=(640)/(-80)→a3/a2=-8
a4/a3=(-5120)/(640)→a4/a3=-8

Then:
(an)/(an-1)=-8

Solving for an:
an=(-8)(an-1)
an=-8 (an-1)

From the task data 
.

If these four terms form the geometrical sequence, then each next term is obtained from the previous by multiplying by the same number q. 

Let's find q: (you can easily check that q is also and ).

Then then-th term of geometrical sequence may be represented as . That's why .