Find the 6th Term -324 , 108 , -36 , 12

$- 324$ , $108$ , $- 36$ , $12$

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 $- \frac{1}{3}$ gives the next term. In other words, $a_{n} = a_{1} \cdot r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{3}$

This is the form of a geometric sequence.

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

Substitute in the values of $a_{1} = - 324$ and $r = - \frac{1}{3}$ .

$a_{n} = (- 324) \cdot {(- \frac{1}{3})}^{n - 1}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{1}{3}$ .

$a_{n} = - 324 \cdot ({(- 1)}^{n - 1} {(\frac{1}{3})}^{n - 1})$

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = - 324 \cdot ({(- 1)}^{n - 1} (\frac{1^{n - 1}}{3^{n - 1}}))$

One to any power is one.

$a_{n} = - 324 \cdot ({(- 1)}^{n - 1} (\frac{1}{3^{n - 1}}))$

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = - 324 \cdot \frac{{(- 1)}^{n - 1}}{3^{n - 1}}$

Combine $- 324$ and $\frac{{(- 1)}^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{- 324 {(- 1)}^{n - 1}}{3^{n - 1}}$

Move the negative in front of the fraction.

$a_{n} = - \frac{324 {(- 1)}^{n - 1}}{3^{n - 1}}$

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

$a_{6} = - \frac{324 {(- 1)}^{(6) - 1}}{3^{(6) - 1}}$

Simplify the numerator.

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Subtract $1$ from $6$ .

$a_{6} = - \frac{324 {(- 1)}^{5}}{3^{(6) - 1}}$

Raise $- 1$ to the power of $5$ .

$a_{6} = - \frac{324 \cdot - 1}{3^{(6) - 1}}$

Simplify the denominator.

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Subtract $1$ from $6$ .

$a_{6} = - \frac{324 \cdot - 1}{3^{5}}$

Raise $3$ to the power of $5$ .

$a_{6} = - \frac{324 \cdot - 1}{243}$

Reduce the expression by cancelling the common factors.

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Multiply $324$ by $- 1$ .

$a_{6} = - \frac{- 324}{243}$

Cancel the common factor of $- 324$ and $243$ .

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Factor $81$ out of $- 324$ .

$a_{6} = - \frac{81 (- 4)}{243}$

Cancel the common factors.

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Factor $81$ out of $243$ .

$a_{6} = - \frac{81 \cdot - 4}{81 \cdot 3}$

Cancel the common factor.

$a_{6} = - \frac{81 \cdot - 4}{81 \cdot 3}$

Rewrite the expression.

$a_{6} = - \frac{- 4}{3}$

Move the negative in front of the fraction.

$a_{6} = \frac{4}{3}$

Find the 6th Term -324 , 108 , -36 , 12

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