find the common ratio $r$ for the geometric sequence and use $r$ to find the next three terms.

2. $972, 324, 108, 36, \dots$

first, identify each term in the sequence. for this sequence,

$f(1) = 972$, $f(2) = 324$, $f(3) = 108$, and $f(4) = 36$

then, find the common ratio $r$.

$r = \frac{324}{972} = \frac{1}{3}$

lastly, use $f(n) = f(n - 1) \cdot r$ with $r = \text{___}$ to find $f(5)$, $f(6)$, and $f(7)$.

$f(5) = f(4) \cdot \text{___}$
$f(5) = \text{___} \cdot \text{___}$
$f(5) = \text{___}$

$f(6) = f(5) \cdot \text{___}$
$f(6) = \text{___} \cdot \text{___}$
$f(6) = \text{___}$

$f(7) = f(6) \cdot \text{___}$
$f(7) = \text{___} \cdot \text{___}$
$f(7) = \text{___}$

the common ratio of the geometric sequence is $\text{___}$ and the next three terms of the sequence are $\text{___}$, $\text{___}$, and $\text{___}$.

Question

find the common ratio $r$ for the geometric sequence and use $r$ to find the next three terms.

2. $972, 324, 108, 36, \dots$

first, identify each term in the sequence. for this sequence,

$f(1) = 972$, $f(2) = 324$, $f(3) = 108$, and $f(4) = 36$

then, find the common ratio $r$.

$r = \frac{324}{972} = \frac{1}{3}$

lastly, use $f(n) = f(n - 1) \cdot r$ with $r = \text{___}$ to find $f(5)$, $f(6)$, and $f(7)$.

$f(5) = f(4) \cdot \text{___}$
$f(5) = \text{___} \cdot \text{___}$
$f(5) = \text{___}$

$f(6) = f(5) \cdot \text{___}$
$f(6) = \text{___} \cdot \text{___}$
$f(6) = \text{___}$

$f(7) = f(6) \cdot \text{___}$
$f(7) = \text{___} \cdot \text{___}$
$f(7) = \text{___}$

the common ratio of the geometric sequence is $\text{___}$ and the next three terms of the sequence are $\text{___}$, $\text{___}$, and $\text{___}$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Geometric Sequences", "Common Ratio" ], "new_concepts": [], "current_concepts": [ "Geometric Sequences", "Common Ratio" ] } ## Find the common ratio Using the *Common Ratio* knowledge point $$ r = \frac{f(2)}{f(1)} = \frac{324}{972} = \frac{1}{3} $$ ## Calculate the fifth term Using the *Geometric Sequences* knowledge point $$ \begin{aligned} f(5) &= f(4) \cdot r \ f(5) &= 36 \cdot \frac{1}{3} \ f(5) &= 12 \end{aligned} $$ ## Calculate the sixth term Using the *Geometric Sequences* knowledge point $$ \begin{aligned} f(6) &= f(5) \cdot r \ f(6) &= 12 \cdot \frac{1}{3} \ f(6) &= 4 \end{aligned} $$ ## Calculate the seventh term Using the *Geometric Sequences* knowledge point $$ \begin{aligned} f(7) &= f(6) \cdot r \ f(7) &= 4 \cdot \frac{1}{3} \ f(7) &= \frac{4}{3} \end{aligned} $$ Find the common ratio $r$ for the geometric sequence and use $r$ to find the next three terms. 2. $972, 324, 108, 36, \dots$ First, identify each term in the sequence. For this sequence, $$f(1) = 972, f(2) = 324, f(3) = 108, \text{ and } f(4) = 36$$ Then, find the common ratio $r$. $$r = \frac{324}{972} = \frac{1}{3}$$ Lastly, use $f(n) = f(n - 1) \cdot r$ with $r =$ $\frac{1}{3}$ to find $f(5)$, $f(6)$, and $f(7)$. $$f(5) = f(4) \cdot \text{}$\frac{1}{3}$$$ $$f(5) = \text{}36 \cdot \text{}$\frac{1}{3}$$$ $$f(5) = \text{}12$$ $$f(6) = f(5) \cdot \text{}$\frac{1}{3}$$$ $$f(6) = \text{}12 \cdot \text{}$\frac{1}{3}$$$ $$f(6) = \text{}4$$ $$f(7) = f(6) \cdot \text{}$\frac{1}{3}$$$ $$f(7) = \text{}4 \cdot \text{}$\frac{1}{3}$$$ $$f(7) = \text{}$\frac{4}{3}$$$ The common ratio of the geometric sequence is $\frac{1}{3}$ and the next three terms of the sequence are 12, 4, and $\frac{4}{3}$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Geometric Sequences" ] }

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QUESTION IMAGE

Question

Explanation:

Find the common ratio

Calculate the fifth term

Calculate the sixth term

Calculate the seventh term

Answer: