Question

What is the eighth term in the sequence x+3, - 2x^2 - 6x, 4x^3 +12x^2​

Answer 1

`a_8=-128x^8-384x^7`

Explanation:

The terms of the sequence are:

`x+3,-2x^2-6x,4x^3+12x^2,...`

We can rewrite the terms in factored form to get;

`x+3,-2x(x+3),4x^2(x+3),...`

We can see that the subsequent terms are obtained by multiplying the previous term by
`-2x`. This is called the common ratio.

Therefore the first term of this geometric sequence is
`a_1=x+3` and the common ratio is
`r=-2x`.

The nth term of a geometric sequence is given by:
`a_n=a_1(r^(n-1))`.

Let us substitute the first term, the common ratio, and
`n=8` to obtain:

`a_8=(x+3)(-2x)^(8-1)`

`a_8=(x+3)(-2x)^(7)`

`a_8=-128x^7(x+3)`

`a_8=-128x^8-384x^7`