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In a geometric sequence, the first term,
a_(1), is equal to 1 , and the fourth term,
a_(4), is equal to 64 . Which number represents the common ratio of the geometric sequence?

r=2

r=3

r=4

r=5

In a geometric sequence, the first term, $a_{1}$ , is equal to $1$ , and the fourth term, $a_{4}$ , is equal to $64$ . Which number represents the common ratio of the geometric sequence? $\newline$ $r=2$ $\newline$ $r=3$ $\newline$ $r=4$ $\newline$ $r=5$

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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 fourth term,

a_{4}

, is equal to

64

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

\newline

r=2

\newline

r=3

\newline

r=4

\newline

r=5

Identify Given Terms: Identify the given terms in the geometric sequence. $\newline$ We are given the first term $a_{1}$ and the fourth term $a_{4}$ of the geometric sequence. The first term is $1$ $a_{1} = 1$ and the fourth term is $64$ $a_{4} = 64$ .
Recall Formula: Recall the formula for the $n$ th term of a geometric sequence. $\newline$ The $n$ th term of a geometric sequence can be found using the formula $a_n = a_1 \cdot r^{(n-1)}$ , where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.
Set Up Equation: Set up the equation to find the common ratio using the given terms. $\newline$ We know that $a_{4} = a_{1} \times r^{4-1}$ , which simplifies to $64 = 1 \times r^3$ .
Solve for Ratio: Solve for the common ratio $r$ . Since $a_{1} = 1$ , the equation simplifies to $64 = r^3$ . To find $r$ , we need to take the cube root of both sides of the equation. The cube root of $64$ is $4$ , so $r = 4$ .

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