Answer:
[tex]a_8=-128x^8-384x^7[/tex]
Step-by-step explanation:
The terms of the sequence are:
[tex]x+3,-2x^2-6x,4x^3+12x^2,...[/tex]
We can rewrite the terms in factored form to get;
[tex]x+3,-2x(x+3),4x^2(x+3),...[/tex]
We can see that the subsequent terms are obtained by multiplying the previous term by [tex]-2x[/tex]. This is called the common ratio.
Therefore the first term of this geometric sequence is [tex]a_1=x+3[/tex] and the common ratio is [tex]r=-2x[/tex].
The nth term of a geometric sequence is given by: [tex]a_n=a_1(r^{n-1})[/tex].
Let us substitute the first term, the common ratio, and [tex]n=8[/tex] to obtain:
[tex]a_8=(x+3)(-2x)^{8-1}[/tex]
[tex]a_8=(x+3)(-2x)^{7}[/tex]
[tex]a_8=-128x^7(x+3)[/tex]
[tex]a_8=-128x^8-384x^7[/tex]
