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?
○ -128x⁸ - 384x⁷
○ 128x⁸ + 384x⁷
○ 256x⁹ + 768x⁸
○ -256x⁹ - 768x⁸

Explanation:

Step1: Find the common ratio \( r \)

In a geometric sequence, the common ratio \( r \) is the ratio of any term to its previous term. Let's take the second term divided by the first term:
First term \( a_1 = x + 3 \)
Second term \( a_2=-2x^2 - 6x=-2x(x + 3) \)
So \( r=\frac{a_2}{a_1}=\frac{-2x(x + 3)}{x + 3}=-2x \) (assuming \( x
eq - 3 \))
We can verify with the third term and the second term:
Third term \( a_3 = 4x^3+12x^2 = 4x^2(x + 3) \)
\( \frac{a_3}{a_2}=\frac{4x^2(x + 3)}{-2x(x + 3)}=-2x \), which matches the common ratio we found.

Step2: Recall the formula for the \( n \)-th term of a geometric sequence

The formula for the \( n \)-th term of a geometric sequence is \( a_n=a_1\times r^{n - 1} \)

Step3: Identify \( a_1 \), \( r \) and \( n \) for the eighth term

We have \( a_1=x + 3 \), \( r=-2x \), and \( n = 8 \)

Step4: Substitute into the formula

\( a_8=(x + 3)\times(-2x)^{8 - 1}=(x + 3)\times(-2x)^7 \)
First, calculate \( (-2x)^7=(-2)^7\times x^7=-128x^7 \)
Then, \( a_8=(x + 3)\times(-128x^7)=-128x^7\times x-128x^7\times3=-128x^8-384x^7 \)

Answer:

\( -128x^8 - 384x^7 \)