`a_1 = 80, a_(k + 1) = (-1/2)a_k` Write the first five terms of the geometric sequence. Determine the common ratio and write the nth term of...

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,...

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)` .