Question

The first three terms of a geometric sequence are shown below. x+3,-2x2-6x,4x3+12x2,.... What is the eighth term of the sequence?

Answer 1

You have first three terms of the geometrical sequence:

$a_1=x+3,\\ a_2=-2x^2-6x=-2x(x+3),\\a_3=4x^3+12x^2=4x^2(x+3).$

You should know how the terms of geometrical sequence are formed:

$a_2=a_1\cdot r,\\ a_3=a_2\cdot r=a_1\cdot r^2,...\\a_n=a_1\cdot r^{n-1}$

where r is a ratio of geometrical sequence.

Comparing first three given terms, you can conclude that $r=-2x$ and $a_1=x+3$. Then

$a_8=a_1\cdot r^7=(x+3)\cdot (-2x)^7=-128x^7(x+3).$

Answer: $a_8=-128x^7(x+3).$

Answer 2

You can use the fact that a geometric series has two adjacent terms differing by a fixed constant.

The eighth term of the considered geometric sequence is given by

$-128x^8 - 384x^7$

What is  a geometric sequence and how to find its nth terms?

There are three parameters which differentiate between which geometric sequence we're taking about.

The first parameter is the initial value of the sequence.

The second parameter is the quantity by which we multiply previous term to get the next term.

The third parameter is the length of the sequence. It can be finite or infinite.

Suppose the initial term of a geometric sequence is $a$   and the term by which we multiply the previous term to get the next term is  $d$

Then the sequence would look like

$a, ad, ad^2, ad^3, ...$ (till the terms to which it is defined)

Thus, the nth term of such sequence would be

$ad^{n-1}$

How to find the eighth term of the sequence?

First we will have to find the factor by which previous terms are multiplied to obtain the next term. And we can then try relating it with its terms to find the eighth term.

Since it is given that the given sequence is geometric sequence, thus, we have

$T_2 = d \times T_1\\\\d = \dfrac{T_2}{T_1}$

This will help find the factor d.

From the given sequence, we have

$T_1 = x + 3\\T_2 = -2x^2 - 6x$

Thus,

$d = \dfrac{-2x^2 - 6x}{x+3} = \dfrac{-2x(x+3)}{x+3} = -2x$

Thus, the eighth term is obtained by

$T_8 = T_1 \times d^{8-1} = T_1d^7\\\\T_8 = (x+3){-2x}^7 = (x+3)(-128x^7) = -128x^8 - 384x^7$

Thus, the eighth term in the given sequence is

$-128x^8 - 384x^7$

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