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In a geometric sequence, the first term,
a_(1), is equal to 3 , and the third term,
a_(3), is equal to 192 . Which number represents the common ratio of the geometric sequence?

r=6

r=7

r=8

r=9

In a geometric sequence, the first term, $a_{1}$ , is equal to $3$ , and the third term, $a_{3}$ , is equal to $192$ . Which number represents the common ratio of the geometric sequence? $\newline$ $r=6$ $\newline$ $r=7$ $\newline$ $r=8$ $\newline$ $r=9$

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Evaluate two-variable equations: word problems

Full solution

Q. In a geometric sequence, the first term,

a_{1}

, is equal to

3

, and the third term,

a_{3}

, is equal to

192

. Which number represents the common ratio of the geometric sequence?

\newline

r=6

\newline

r=7

\newline

r=8

\newline

r=9

Identify Given Terms: Identify the given terms in the geometric sequence. We are given the first term $a_{1} = 3$ and the third term $a_{3} = 192$ . In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio ( $r$ ).
Write Third Term Formula: Write the formula for the third term of a geometric sequence. $\newline$ The $n$ th term of a geometric sequence can be found using the formula $a_{n} = a_{1} \cdot r^{n-1}$ . For the third term, the formula is $a_{3} = a_{1} \cdot r^{3-1} = a_{1} \cdot r^2$ .
Substitute Known Values: Substitute the known values into the formula. $\newline$ We know that $a_{1} = 3$ and $a_{3} = 192$ , so we can substitute these values into the formula to find $r^2$ . $\newline$ $192 = 3 \times r^2$
Solve for $r^2$ : Solve for $r^2$ . $\newline$ To find $r^2$ , we divide both sides of the equation by $3$ . $\newline$ $r^2 = \frac{192}{3}$ $\newline$ $r^2 = 64$
Find Value of r: Find the value of r. $\newline$ Since $r^2 = 64$ , we take the square root of both sides to solve for r. $\newline$ $r = \sqrt{64}$ $\newline$ $r = 8$ or $r = -8$
Determine Appropriate Value: Determine the appropriate value of $r$ . In the context of a geometric sequence, the common ratio can be positive or negative. However, since we are given options for $r$ and they are all positive, we choose $r = 8$ as the common ratio.

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