Question

Enter a recursive rule for the geometric sequence.

Answer 1

From the task data 
$a_1=10 \\ a_2=-80 \\ a_3=640 \\ a_4=-5120$.

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:$a_2=a_1\cdot q \Rightarrow q= \frac{a_2}{a_1}= \frac{-80}{10} =-8$ (you can easily check that q is also $\frac{a_3}{a_2}$ and $\frac{a_4}{a_3}$ ).

Then then-th term of geometrical sequence may be represented as $a_n=a_1q^{n-1}$. That's why $a_n=10\cdot (-8)^{n-1}$.

Answer 2

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)