Find the 5th Term 243 , 162 , 108 , 72

$243$ , $162$ , $108$ , $72$

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 $a_{1} = 243$ and $r = \frac{2}{3}$ .

$a_{n} = 243 {(\frac{2}{3})}^{n - 1}$

Step 4

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

$a_{n} = 243 \frac{2^{n - 1}}{3^{n - 1}}$

Step 5

Combine $243$ and $\frac{2^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{243 \cdot 2^{n - 1}}{3^{n - 1}}$

Step 6

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

$a_{5} = \frac{243 \cdot 2^{(5) - 1}}{3^{(5) - 1}}$

Step 7

Simplify the numerator.

Subtract $1$ from $5$ .

$a_{5} = \frac{243 \cdot 2^{4}}{3^{5 - 1}}$

Raise $2$ to the power of $4$ .

$a_{5} = \frac{243 \cdot 16}{3^{5 - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $5$ .

$a_{5} = \frac{243 \cdot 16}{3^{4}}$

Raise $3$ to the power of $4$ .

$a_{5} = \frac{243 \cdot 16}{81}$

Step 9

Simplify the expression.

Multiply $243$ by $16$ .

$a_{5} = \frac{3888}{81}$

Divide $3888$ by $81$ .

$a_{5} = 48$

Find the 5th Term 243 , 162 , 108 , 72

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