Question

Enter a recursive rule for the geometric sequence.

10,−80,640,−5120,...

a1=
; an=

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= (a_2)/(a_1)= (-80)/(10) =-8` (you can easily check that q is also
`(a_3)/(a_2)` and
`(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)