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In a geometric sequence, the first term,
a_(1), is equal to 6 , and the third term,
a_(3), is equal to 96 . Which number represents the common ratio of the geometric sequence?

r=3

r=4

r=5

r=6

In a geometric sequence, the first term, $a_{1}$ , is equal to $6$ , and the third term, $a_{3}$ , is equal to $96$ . Which number represents the common ratio of the geometric sequence? $\newline$ $r=3$ $\newline$ $r=4$ $\newline$ $r=5$ $\newline$ $r=6$

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Find the sum of a finite geometric series

Full solution

Q. In a geometric sequence, the first term,

a_{1}

, is equal to

6

, and the third term,

a_{3}

, is equal to

96

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

\newline

r=3

\newline

r=4

\newline

r=5

\newline

r=6

Identify Given Terms: Identify the given terms in the geometric sequence. $\newline$ We are given the first term $a_{1}$ as $6$ and the third term $a_{3}$ as $96$ . In a geometric sequence, each term is found by multiplying the previous term by the common ratio $r$ .
Write Formula for Third Term: Write the formula for the third term of a geometric sequence. $\newline$ The third term $a_{3}$ can be expressed in terms of the first term $a_{1}$ and the common ratio $r$ as follows: $\newline$ $a_{3} = a_{1} \cdot r^{2}$
Substitute Known Values: Substitute the known values into the formula. $\newline$ We know that $a_{1} = 6$ and $a_{3} = 96$ , so we can substitute these values into the formula: $\newline$ $96 = 6 \times r^2$
Solve for Common Ratio: Solve for the common ratio $r$ . To find $r$ , we need to divide both sides of the equation by $6$ : $r^2 = \frac{96}{6}$ $r^2 = 16$
Take Square Root: Take the square root of both sides to solve for $r$ . Since $r^2 = 16$ , we find that $r$ can be either positive or negative square root of $16$ . However, since a common ratio is typically positive in the context of geometric sequences, we will consider the positive square root: $r = \sqrt{16}$ $r = 4$

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