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In a geometric sequence, the first term,
a_(1), is equal to 2 , and the third term,
a_(3), is equal to 72 . 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 $2$ , and the third term, $a_{3}$ , is equal to $72$ . 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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Find the sum of a finite geometric series

Full solution

Q. In a geometric sequence, the first term,

a_{1}

, is equal to

2

, and the third term,

a_{3}

, is equal to

72

. 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. $\newline$ We are given the first term $a_{1}$ as $2$ and the third term $a_{3}$ as $72$ . 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 $a_{1} = 2$ and $a_{3} = 72$ , so we can substitute these values into the formula: $\newline$ $72 = 2 \times r^{2}$
Solve for Common Ratio: Solve for the common ratio $r$ . To find $r$ , we need to divide both sides of the equation by $2$ : $72 / 2 = r^2$ $36 = r^2$ Now, take the square root of both sides to solve for $r$ : $\sqrt{36} = r$ $r = 6$

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