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In a geometric sequence, the first term,
a_(1), is equal to 3 , and the fifth term,
a_(5), is equal to 48 . Which number represents the common ratio of the geometric sequence?

r=1

r=2

r=3

r=4

In a geometric sequence, the first term, $a_{1}$ , is equal to $3$ , and the fifth term, $a_{5}$ , is equal to $48$ . Which number represents the common ratio of the geometric sequence? $\newline$ $r=1$ $\newline$ $r=2$ $\newline$ $r=3$ $\newline$ $r=4$

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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

3

, and the fifth term,

a_{5}

, is equal to

48

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

\newline

r=1

\newline

r=2

\newline

r=3

\newline

r=4

Identify Formula for nth Term: Identify the formula for the nth term of a geometric sequence. The nth term of a geometric sequence can be found using the formula $a_n = a_1 \times r^{(n-1)}$ , where $a_n$ is the nth term, $a_1$ is the first term, and $r$ is the common ratio.
Set Up Equation: Set up the equation using the given terms. $\newline$ We know that $a_1 = 3$ and $a_5 = 48$ . We can use the formula from Step $1$ to write the equation for the fifth term: $48 = 3 \cdot r^{5-1}$ .
Simplify Equation: Simplify the equation. $\newline$ $48 = 3 \times r^4$ $\newline$ Now, we need to solve for $r$ .
Divide to Isolate $r^4$ : Divide both sides of the equation by $3$ to isolate $r^4$ . $\newline$ $\frac{48}{3} = r^4$ $\newline$ $16 = r^4$
Take Fourth Root to Solve: Take the fourth root of both sides to solve for $r$ . $r = 16^{1/4}$ $r = 2$

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