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

$x + 3, -2x^2 - 6x, 4x^3 + 12x^2, \dots$

what is the eighth term of the sequence?

$-128x^8 - 384x^7$
$128x^8 + 384x^7$
$256x^9 + 768x^8$
$-256x^9 - 768x^8$

Question

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

$x + 3, -2x^2 - 6x, 4x^3 + 12x^2, \dots$

what is the eighth term of the sequence?

$-128x^8 - 384x^7$
$128x^8 + 384x^7$
$256x^9 + 768x^8$
$-256x^9 - 768x^8$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Sequences", "Common Ratio", "Geometric Sequence Explicit Formula" ], "new_concepts": [], "current_concepts": [ "Geometric Sequences", "Common Ratio", "Geometric Sequence Explicit Formula" ] } ## Find the common ratio Using the *Common Ratio* knowledge point $$ \begin{aligned} a_1 &= x + 3 \ a_2 &= -2x^2 - 6x = -2x(x + 3) \ r &= \frac{a_2}{a_1} = \frac{-2x(x + 3)}{x + 3} = -2x \end{aligned} $$ ## Verify with the third term Using the *Geometric Sequences* knowledge point $$ \begin{aligned} a_3 &= a_2 \cdot r = -2x(x + 3) \cdot (-2x) = 4x^2(x + 3) = 4x^3 + 12x^2 \end{aligned} $$ ## Find the eighth term Using the *Geometric Sequence Explicit Formula* knowledge point $$ \begin{aligned} a_n &= a_1 \cdot r^{n-1} \ a_8 &= (x + 3) \cdot (-2x)^{7} \ a_8 &= (x + 3) \cdot (-128x^7) \ a_8 &= -128x^8 - 384x^7 \end{aligned} $$

(A) $-128x^8 - 384x^7$

(B) $128x^8 + 384x^7$

(C) $256x^9 + 768x^8$

(D) $-256x^9 - 768x^8$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Sequences" ] }

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QUESTION IMAGE

Question

Explanation:

Find the common ratio

Verify with the third term

Find the eighth term

Answer: