Question

The first three terms of a geometric sequence are shown below.

What is the eighth term of the sequence?
A. -128x^8-384x^7
B. 128x^8+384x^7
C. 256x^9+768x^8
D. -256x^9-768x^8

Answer 1

Each term oscillates between positive and negative. This always means there is a term (-1)⁽ⁿ⁺¹⁾ in there so that when n=1 we get (-1)²=1 and for n=2: (-1)3=-1.
This lets the terms switch signs. Next, we see the first term is changing in magnitude by (2⁽ⁿ⁻¹⁾)xⁿ. The last term is changing sign too and its magnitude is changing by 3(2x)⁽ⁿ⁻¹⁾. Putting it together gives:
(-1)⁽ⁿ⁺¹⁾[(2⁽ⁿ⁻¹⁾)xⁿ + 3(2x)⁽ⁿ⁻¹⁾] with n=8,
-1(2⁷x⁸ + 3(2⁷x⁷)) = -128x⁸ - 384x⁷
So A

Answer 2

Option A is correct.

Explanation:

Given:

First term, a = x + 3

second term = -2x² - 6x

Third term = 4x³ + 12x²

Common ratio, r =
`(-2x^2-6x)/(x+3)=(-2x(x+3))/(x+3)=-2x`

nth term of GP is gievn by,
`a_n=ar^(n-1)`

So, Eight term =
`a_8=(x+3)(-2x)^(8-1)=(x+3)(-2x)^(7)=-128x^8-384x^7`

Therefore, Option A is correct.