Find the 4th Term 54 , 36 , 24

$54$ , $36$ , $24$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $\frac{2}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{2}{3}$

Step 2

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 3

Substitute in the values of $a_{1} = 54$ and $r = \frac{2}{3}$ .

$a_{n} = 54 {(\frac{2}{3})}^{n - 1}$

Step 4

Apply the product rule to $\frac{2}{3}$ .

$a_{n} = 54 \frac{2^{n - 1}}{3^{n - 1}}$

Step 5

Combine $54$ and $\frac{2^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{54 \cdot 2^{n - 1}}{3^{n - 1}}$

Step 6

Substitute in the value of $n$ to find the $n$ th term.

$a_{4} = \frac{54 \cdot 2^{(4) - 1}}{3^{(4) - 1}}$

Step 7

Simplify the numerator.

Subtract $1$ from $4$ .

$a_{4} = \frac{54 \cdot 2^{3}}{3^{4 - 1}}$

Raise $2$ to the power of $3$ .

$a_{4} = \frac{54 \cdot 8}{3^{4 - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $4$ .

$a_{4} = \frac{54 \cdot 8}{3^{3}}$

Raise $3$ to the power of $3$ .

$a_{4} = \frac{54 \cdot 8}{27}$

Step 9

Simplify the expression.

Multiply $54$ by $8$ .

$a_{4} = \frac{432}{27}$

Divide $432$ by $27$ .

$a_{4} = 16$

Find the 4th Term 54 , 36 , 24

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