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In a geometric sequence, the first term,
a_(1), is equal to 1 , and the third term,
a_(3), is equal to 36 . 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 $1$ , and the third term, $a_{3}$ , is equal to $36$ . 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

1

, and the third term,

a_{3}

, is equal to

36

. 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 $a_{1}$ and the third term $a_{3}$ of the geometric sequence. The first term is $1$ $a_{1} = 1$ and the third term is $36$ $a_{3} = 36$ .
Write nth Term Formula: Write the formula for the nth term of a geometric sequence. $\newline$ The nth term of a geometric sequence is given by $a_n = a_1 \cdot r^{(n-1)}$ , where $a_1$ is the first term and $r$ is the common ratio.
Express Third Term: Use the formula to express the third term in terms of the first term and the common ratio. $\newline$ We have $a_{3} = a_{1} \cdot r^{3-1} = a_{1} \cdot r^2$ .
Substitute Given Values: Substitute the given values into the equation. $\newline$ We know that $a_{1} = 1$ and $a_{3} = 36$ , so we can write $36 = 1 \times r^{2}$ .
Solve for Common Ratio: Solve for the common ratio $r$ . To find $r$ , we take the square root of both sides of the equation: $r^2 = 36$ , so $r = \pm\sqrt{36}$ .
Determine Positive Ratio: Determine the positive value of the common ratio. $\newline$ Since a common ratio in a geometric sequence is typically positive, we take the positive square root of $36$ , which is $6$ . Therefore, $r = 6$ .

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