Find the Next Term 9 , 6 , 4

$9$ , $6$ , $4$

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 $a_{1} = 9$ and $r = \frac{2}{3}$ .

$a_{n} = 9 {(\frac{2}{3})}^{n - 1}$

Step 4

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

$a_{n} = 9 \frac{2^{n - 1}}{3^{n - 1}}$

Step 5

Combine $9$ and $\frac{2^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{9 \cdot 2^{n - 1}}{3^{n - 1}}$

Step 6

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

$a_{4} = \frac{9 \cdot 2^{(4) - 1}}{3^{(4) - 1}}$

Step 7

Simplify the numerator.

Subtract $1$ from $4$ .

$a_{4} = \frac{9 \cdot 2^{3}}{3^{4 - 1}}$

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

$a_{4} = \frac{9 \cdot 8}{3^{4 - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $4$ .

$a_{4} = \frac{9 \cdot 8}{3^{3}}$

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

$a_{4} = \frac{9 \cdot 8}{27}$

Step 9

Reduce the expression by cancelling the common factors.

Multiply $9$ by $8$ .

$a_{4} = \frac{72}{27}$

Cancel the common factor of $72$ and $27$ .

Factor $9$ out of $72$ .

$a_{4} = \frac{9 (8)}{27}$

Cancel the common factors.

Factor $9$ out of $27$ .

$a_{4} = \frac{9 \cdot 8}{9 \cdot 3}$

Cancel the common factor.

$a_{4} = \frac{9 \cdot 8}{9 \cdot 3}$

Rewrite the expression.

$a_{4} = \frac{8}{3}$

Find the Next Term 9 , 6 , 4

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