Precalculus, Chapter 9, 9.3, Section 9.3, Problem 28

The given are:
a_1=80
a_(k+1) = (-1/2)a_k
To determine the first five terms of the geometric sequence, plug-in the values k=1,2,3,4 to the given recursive formula.
When k=1, the nth term is:
a_(1+1)=(-1/2)a_1
a_2=(-1/2)*80
a_2=-40
When k=2, the nth term is:
a_(2+1)=(-1/2)a_2
a_3=(-1/2)*(-40)
a_3=20
When k=3, the nth term is:
a_(3+1)=(-1/2)a_3
a_4=(-1/2)*20
a_4=-10
And when k=4, the nth term is:
a_(4+1)=(-1/2)a_4
a_5=(-1/2)*(-10)
a_5=5
Therefore, the first five terms of the geometric sequence are {80, -40, 20, -10, 5} .

To determine the common ratio, apply the formula:
r=a_(n+1)/a_n
So the ratio of the consecutive terms of the geometric sequence is:
r=a_5/a_4=5/(-10)=-1/2
r=a_4/a_3=(-10)/20=-1/2
r=a_3/a_2=20/(-40)=-1/2
r=a_2/a_1=(-40)/80=-1/2
Thus, the common ratio of the geometric sequence is -1/2 .

To determine the nth term of geometric sequence, apply the formula:
a_n=a_1*r^(n-1)
Plugging in the values of a1 and r, the formula becomes:
a_n=80*(-1/2)^(n-1)
Hence, the nth term rule of this geometric sequence is a_n=80*(-1/2)^(n-1) .