Q7. - Algebra 1

Question

Suppose the first term of a geometric sequence is 3 and the fourth term is 192.

a. What is the common ratio of the sequence?

b. Write an equation that can be used to find the $n^{\text{th}}$ term of the sequence.

c. What is the sixth term of the sequence?

Step-by-Step Solution

Answer

a. The common ratio is 4.

b. The equation used to find the $n^{\mathrm{th}}$ term of the sequence is $a_n=3{\left(4\right)}^{n-1}$.

c. The sixth term of the sequence is 3072.

Part a. Step 1. Define a geometric sequence.

A geometric sequence is a sequence of non-zero numbers where each term after the first term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Part a. Step 2. Define the n th term of a geometric sequence.

The $n^{th}$ term of a geometric sequence is given by

$a_n=a_1{r}^{n-1}$

Where,

$a_n=$ The $n^{\mathrm{th}}$ term of the sequence

$a_1=$ The first term of the geometric sequence

$r=$ The common ratio

Part a. Step 3. Determine the common ratio of the given geometric sequence.

The first term of the sequence, $a_1=3$

The fourth term of the sequence, $a_4=192$

Substitute $n=4$ in $a_n=a_1{r}^{n-1}$.

$\begin{array}{c}{a}_4=a_1{r}^{4-1}\\ a_4=a_1{r}^3\end{array}$

Substitute $a_1=3$ and $a_4=192$ in $a_4=a_1{r}^3$.

$$\begin{gathered} 192=3r^3 \\ r^3=\frac{192}{3} \\ r^3=64 \\ r=\sqrt[3]{64} \\ r=4 \end{gathered}$$

Hence, the common ratio is 4.

Part b. Step 1. Define a geometric sequence.

A geometric sequence is a sequence of non-zero numbers where each term after the first term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Part b. Step 2. Define the n th term of a geometric sequence.

The $n^{\mathrm{th}}$ term of a geometric sequence is given by

$a_n=a_1{r}^{n-1}$

Where, 

$a_n=$ The $n^{\mathrm{th}}$ term of the sequence

$a_1=$ The first term of the geometric sequence

$r=$ The common ratio

Part b. Step 3. Determine the common ratio of the given geometric sequence.

Substitute $a_1=3$ and $r=4$ in $a_n=a_1{r}^{n-1}$.

$a_n=3{\left(4\right)}^{n-1}$

Hence, the equation used to find the $n^{\mathrm{th}}$ term of the sequence is $a_n=3{\left(4\right)}^{n-1}$.

Part c. Step 1. Define a geometric sequence.

A geometric sequence is a sequence of non-zero numbers where each term after the first term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Part c. Step 2. Define the term of a geometric sequence.

The $n^{\mathrm{th}}$ term of a geometric sequence is given by

$a_n=a_1{r}^{n-1}$

Where, 

$a_n=$ The width="21" height="22" style="max-width: none; vertical-align: -4px;" $n^{\mathrm{th}}$ term of the sequence

$a_1=$ The first term of the geometric sequence

$r=$ The common ratio

Part c. Step 3. Determine the common ratio of the given geometric sequence.

Substitute $n=6$, $a_1=3$ and $r=4$ in $a_n=a_1{r}^{n-1}$.

$$\begin{gathered} a_6=3{\left(4\right)}^{6-1} \\ {a}_6=3{\left(4\right)}^5 \\ {a}_6=3\left(1024\right) \\ {a}_6=3072 \end{gathered}$$

Hence, the sixth term of the sequence is 3072.